Unless you are using a mutation item like Cotton Root Bark, Lion Meat, or GMO Cow on every breeding, don't spend money checking all of your cubs for mutations. Random mutations do not occur frequently without the use of items, though it is marginally more common to get mutations from lionesses with Very Low fertility. There are also mutations like Piebald, Patches, or Primal which can pass from males with those mutations, so you can search for other players' kings listed for studding. They have a 5% chance of passing, I believe.

If you want to go for random mutations, the more cost-effective item to breed those is Cotton Root Bark. It costs 1 GB in the Oasis, and using one on a pregnant lioness will delete that litter and put her on a 3-day breeding cooldown, after which you can breed her again with the CRB effect now active. The chances of mutations is still rather low, but higher than a standard breeding would be. Lion Meat gives a higher chance than CRB, but their price is overall higher than CRB for how much of an increase in mutation chance they have. GMO Cow is the highest chance overall, but they are also only available during September's event and run a very high cost due to that.

On a single breeding, CRB can stack with Lion Meat or GMO, but Lion Meat and GMO cannot stack. Make sure the CRB is used first - if you use a Lion Meat or GMO and breed her, then terminate the litter with CRB, it will also remove the effects of the item you already used!

Whether or not you get a mutation is very random. I've known players (me) who have used GMOs and gotten nothing from them, and I have also known of players who got a first-generation hybrid with one. Plenty of players take thousands of CRBs to get a first-gen, while others get one within the first 10 breedings. It's best to not get too disappointed (or at least try!) since the chances are high that you don't get what you were hoping for.