Counting polyominoes on (hyper-)cubes
Python 3, 389

import itertools as I
S=sorted
P=I.product
def C(n,m,k):
 Q=[((-1,)*(n-m)+(0,)*m,)]
 for i in' '*(k-1):Q=set(tuple(S(q+(v,)))for q in Q for v in P(*[(-1,0,1)]*n)if sum(map(abs,v))==n-m if not v in q and any(sum((a!=b)*(1+2*a*b)for a,b in zip(v,u))==2for u in q))
 return sum(all(S(q)&lt;=S(zip(*r))for X in I.permutations(zip(*q))for r in P(*((p,tuple(-x for x in p)) for p in X)))for q in Q)


Try it online!

Basically just finds all connected polyominos, and discards ones which can be rotated into a lexicographically smaller polyomino, with rotations being brute-forced.

Can definitely be improved but it's my bedtime.