  We determine, up to a multiplicative constant, the optimal number of random
edges that need to be added to a $k$-graph $H$ with minimum vertex degree
$\Omega(n^{k-1})$ to ensure an $F$-factor with high probability, for any $F$
that belongs to a certain class $\mathcal{F}$ of $k$-graphs, which includes,
e.g., all $k$-partite $k$-graphs, $K_4^{(3)-}$ and the Fano plane. In
particular, taking $F$ to be a single edge, this settles a problem of
Krivelevich, Kwan and Sudakov [Combin. Probab. Comput. 25 (2016), 909--927]. We
also address the case in which the host graph $H$ is not dense, indicating that
starting from certain such $H$ is essentially the same as starting from an
empty graph (namely, the purely random model).
