You are trying for a shelter adoption, by way of newspaper or computer? There may well be an easier way, but if you are using NRaas mods I would suggest using MasterController to TA (Totally Annihilate) the animals in the adoption pool so that presumably new ones will spawn to take their places. The adoption pool animals are technically Service Sims, so it may take some fiddling around with MC's filters to get them to appear in City Hall's MC selection window.
Another option is to switch active households and get some other family to adopt the poor unloved kitty. Or adopt the cat and either change the cat's traits yourself or give it away.
Personally, I wouldn't have the patience to wait for the luck of the draw to produce the pets I really wanted. If no one in town is hosting a cat giving birth to kittens, I would design my own animals, place then with other households moving into town (temporarily if I didn't really want new households coming in) and shift the pet into my own household either by friendly social interactions or by using MasterController's Add Sim.