$H$ is a supergraph of $G$ if it can be obtained from $G$ by adding, edges, vertices, or both.
Is there a standard terminology for a graph that can be obtained from $G$ only by adding edges?
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In graph modification algorithms, a graph $H$ is called a completion of $G$ if there exists a set of edges $F \subseteq [V(G)]^2$ such that $G+F \cong H$.
You can search on arxiv for examples such as
We often think of a graph as the set of its edges, the vertex set being fixed. This is not a universal point of view, but is common enough. In that case you can just write $H \supseteq G$, and say that $H$ is a supergraph of $G$. Just make sure that you explain what you mean by these notations.
Generally speaking, subgraph is ambiguous — it's not clear whether you're allowed to remove vertices. Supergraph is somewhat less common, and suffers from similar ambiguity. Whenever a notion is ambiguous, and you are worried that the reader won't be able to guess correctly from context, you should explain clearly what you mean, humpty-dumpty style.