Hashing a counter to prevent distinguishers in CTR mode
I'm reading the question as generating a keystream per: $S_i\gets E_K(F(\mathrm{IV},i))$ where $F$ is a public function built from a hash function; for incremental index $i$ starting from $0$ and public random $\mathrm{IV}$, with $F$ having a sizable collision rate when $i$ varies; and for $E$ a block cipher with secret key $K$.

That is quite insecure: the location of the collisions, that is values of $i$, $j$ with $i\ne j$ and $F(\mathrm{IV},i)=F(\mathrm{IV},j)$, can be found by an adversary from the public $\mathrm{IV}$. And then knowledge that $K_i=K_j$ can be used to mount an attack allowing decryption of some plaintext fragment from partially known plaintext.