  Motivated by analogous questions in the setting of Steiner triple systems and
Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems,
Journal of Combinatorial Theory Series B, 2020] recently introduced the notion
of graph deficiency. Given a global spanning property $\mathcal P$ and a graph
$G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the
property $\mathcal P$ is the smallest non-negative integer $t$ such that the
join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and
Wagner raised the question of determining how many edges an $n$-vertex graph
$G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $r\geq 3$).
In this paper we resolve their problem fully. We also give an analogous result
which forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of
bounded degree and small bandwidth.
