What is the meaning of -p?

It can be detected today only in a few obscured derivations such as harap (to bite), állapot (state), csillapodik (to calm down).

It can be analyzed from a few derivatives. Presumably it became obsolete already in the Proto-Hungarian period. Its voiced variant -b has also developed. Found today only in a few words, such as közép, hasáb, nyaláb. When the language innovators created the nouns alap (base) and szerep (role), they inferred the -p suffix from back-formations such as telep back-formed from telepedik, ülep back-formed from ülepedik which were originally formed with the instantaneous suffix -p.

Third-person present singular marker.

alternative form of -up

Genitive marker following vowels. See also: -pa.

Source: wiktionary.org