Does the Catholic Church believe in the existance of neanderthals?
Do you mean their existence in pre-history, or whether they still exist today?
The Catholic Church has no teaching on Neanderthals, yea or nay. It supports the sciences in general and believes that the sciences are one method of obtaining truth on our history. This allows for “belief” that Neanderthals existed and even interbred with some populations of homo sapiens, as recent science has demonstrated after DNA analysis.
Now, as to whether Neanderthals were ensouled individuals is something I’m not quite sure. They’d only be so, of course, if they were also descended from Adam and Eve. That much is dogma. All ensouled humans (true, metaphysical humans declared “very good” by God) are descended from Adam and Eve. They would not be so separate from that.
Their existance in pre-history.
The Church has no teaching on this and you are free to hold that they existed or did not exist. If your deeper question is “does the Church believe in evolution” she also has no teaching on this. You are free to accept or reject it.