What is the name of a rooted tree whose nodes may have edges to their descendants?
I'm not aware of any standard term. The graphs that you describe are subgraphs of the transitive closure of a tree, if that's of any help to you, but one wouldn't want to use that phrase twenty times in a paper.

But... as Pål GD alludes to in a comment, every DAG on $n$&nbsp;vertices is a subgraph of the transitive closure of the $n$-vertex directed path. So this answer is a bit of a fail, really.