At the start of a game of Castles of Mad King Ludwig every player gets 3 utility cards and must choose 2 of them to keep. These cards give bonus points at the end of the game based on the castle you have built. There is a pretty fair range in the quality of the cards, and some are clearly more powerful than others. There are two questions to answer. First, which cards are better in general, and how does the existence of the King's Favour change the value of the cards?
Among the utility cards that give you bonus points based on room size there is serious equality: All of them either give 3 points per room with 6 copies of each room in the game, or they give 2 points per room with 9 copies of the room in the game. In either case you have 18 points available in the game. Taking a bunch of copies of a single size does make it easier to get a bunch of points if you empty out that stack, so these are a little better than the 18 points would indicate.
However, the cards that award points based on room colours are much more imbalanced. When you multiply the card point value by the number of those rooms in the game you get the following amounts:
This means that colour based utility cards are generally about the same quality as size based cards when you account for the 2 points per depleted stack benefit for stacking size, but the purple and yellow cards seem like clearly the top of the heap.
There are also cards for round rooms and square rooms, and they both have a value of 15 which makes them quite weak because they are harder to focus on to generate big numbers. You could potentially combine them with size cards to get a lot of points for each purchase but generally I think they are not good.
The cards that give you points for completed rooms and open entrances seem to cluster really tightly around 4 points per game. You can get higher, but I don't think I have ever seen them go above 6 points. They are extremely reliable, require no investment to pay off, and are a fine fallback.
The three utility cards that give points for various corridor cards are really different from each other. The one that boosts hallways is garbage. It is extremely difficult to get more than 3 points out of it, and mostly you will get 1. The one that gives 1 point for every corridor is okay, almost never exciting, and strictly better than the hallway one for some reason...? The third one that gives 2 points per stairs is tricky. In the late game it is trash. On most normal boards it is trash. But it is possible but quite involved to build tons of stairs and basement rooms completing each other and do really well. If you empty the stairs stack you rack up 4 points for each of them which is decent. However, it is a big investment for a not spectacular return so the card overall isn't great.
One card gives you points for every 5 coins you have left at game end. It isn't great and getting a big score off of it requires you to waste a huge amount of currency. Mediocre at baseline.
The last two cards give you 8 points for having all 10 room sizes or 7 points for having all 8 room types. I don't think either of them is great, but I like the 7 point one for all room types much better. There are at least 9 of every room type and only 6 of each room size for some sizes, so the room type is a much easier condition, plus I find it is actually useful to have all of the room types in your castle whereas having all room sizes has no obvious utility. Both of those cards suffer from the fact that you can't reliably plan around them in the early game because you don't know what you will lack in the endgame. They often revolve around desperately hoping you get to buy a room you don't much want on the last turn and often the people aiming for them fail. I really don't like the idea of a bonus card that you spend resources trying to fulfill and end up scoring 0 points for. Perhaps the best argument against these all or nothing cards is that they combine terribly with the other cards. If you want a bunch of blues, for example, it is going to be extremely difficult to also get the variety needed to fulfill one of these cards.
In summary, I think the strategy for keeping starting utility cards can be summarized by a priority list.
Yellow or Purple Colour 2 cards
Other Colour / Size 16 cards
Completed Rooms / Open Entrances 2 cards
Cash/Round Rooms/Square Rooms/Corridors/All Colours/Stairs 6 cards
All Sizes/Hallways 2 cards
Now I want to consider how we combine this with the King's Favour pucks. Say the King's Favour has the green room puck. Everyone now has an added incentive to build green rooms. If you also have the green room utility card, should you keep it? Generally I see people keeping cards that match the Favours because they think that since they are going to want green cards anyway, might as well score even more for them!
This is not the right approach. The thing about rooms affected by the Favour is that everyone wants them and there is a real incentive to get at least one of them. Having a single Favoured room gives you 1 point at worst and often more than that. Moreover it puts you in contention to jump into first place and score up 8 points if other people get stuck at single Favoured room and you can scoop up a second one. Even if you just tie with one other player at two Favoured rooms each you grab 6 points, so threatening to do that is powerful. It is common for one player to corner a lot of a particular colour or size of room under normal circumstances, but doing so when it is the Favoured one is rare and extremely costly.
The key is to remember that nearly every room will be bought at some point. It isn't as though the ones that aren't worth a ton of points are ones you will just ignore. You will end up with low value rooms and it will be a lot easier to collect a bunch of them if other people aren't aiming for them particularly, especially if they don't feel that they need at least one.
Ideally you want to take utility cards that give bonuses to the rooms that *aren't* part of the Favours. You want to be able to cash in on stuff that other people will hand over for cheap, and you don't want to be hunting for room types that everyone wants one of.
So if you see green rooms in the Favours and you have a green room card and a orange room card, keep the orange. You have a much greater chance of being able to grab a ton of orange and get an outstanding result.
This is especially true because in a four player game you will win more if you take risks. A player who gets 5 guaranteed points will not win as often as a player who takes a series of coin flips for 10 or 0 points. You have to have a big score to beat all three players so fighting for the same thing as everyone else and getting predictable, moderate scores is not the ticket. Aim for something different and try to catch em all.
How much does this matter? 14 of the pucks are coloured ones, and there are 10 other pucks. (I count the corridor pucks among the 'other' because they work so differently from the coloured pucks.) When a coloured puck is out that matches a starting card of yours, I would downgrade the card by one tier. Much of the time this won't matter because even if a blue card is worse when blue pucks are out it is still vastly superior to garbage like the 'collect all' cards or the hallways card. You will probably still have some other trash you can safely dump but if you happen to have 3 similar cards drop the one that matches the Favour.
Strangely I think the corridor Favours work the opposite way. Even if they are in play people aren't going to be spending their turns spamming hallways or stairs, it just isn't efficient. If you happen to have the stairs card or the corridors card when a corridors Favour is out, I think it becomes a really reasonable keep and I would upgrade them both by a tier. You can pretty easily scoop the Favour and a bunch of points for your cards and that could be quite the coup.
The cash Favour works in favour of the cash card, like I said earlier. It incentivizes people to complete greens and keep their money around on their Master Builder turn instead of launching it off to the bank. There will be more money around as people fight for the cash Favour, which helps you get more points off of the cash card even if you don't win the Favour.
The other Favours are broad, so I don't think they have a significant effect on card valuation.
That is quite the wall of text, so I will give a quick summary: Keep the starting utility cards that give bonuses for room colours (except corridor ones) or sizes. If you have to choose between those, keep the ones that are different from the Favours. If you have to choose between lesser cards, keep the ones that match the Favours instead.
*Edit: I changed the advice regarding the Stairs card because I didn't know the rule that you can't attach Stairs to Stairs, which makes a heavy Stairs strategy a lot harder to do. It is still possible, but it is much weaker.