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It's also worth mentioning replacement can fail in $H_{\kappa}.$ For example, in the Feferman-Levy extension of $L,$ $H_{\omega_1}$ fails replacement since the $\omega_n^L$'s are definable in $H_{\omega_1}.$
Alright. Also, I do think there's an interesting question to be asked here if power set is removed, and we look for strengthenings of ZFfin - Power set which interpret $\text{ZFC}^-.$
@AlexMennen Take an infinite set $X.$ Then $\mathcal{P}(\mathcal{P}(X))$ has a countably infinite subset, setting $X_n=\{Y \subset X: |Y|=n\}.$ A replacement on that should yield $\omega.$
