What can I do about a hole for a dryer vent that was cut too big?
A good solution would be to make a collar/cover of a material that would fit visually with the outside wall, and would provide a reliable seal to prevent water, air, and pest infiltration. Then to support a good seal, use spray foam to insulate around the duct.

The appropriate material for the collar depends on the wall you're going through. If it's a masonry wall, you can use mortar or a concrete patching compound. If the outside is vinyl siding, a small sheet of vinyl could work, with the appropriate trim pieces to terminate the cut vinyl boards (you may want to cut them square so this looks better). If the outside is wooden shingles or clapboard, a painted piece of pressure-treated plywood or cellular PVC trim board would look OK.

Depending on what exactly was cut, and how much, you may want to replace some of your sheathing to support the siding or collar pieces.

Make sure to use the appropriate trim and caulking to seal the collar to the exterior wall. This is really important to seal out water. Consider how water would flow onto and off of the surface - you want gravity on your side. So if you're doing this with wood shingles, make sure the shingles overlap your collar piece on top, and your collar overlaps the shingles below. You don't want a situation where if your caulk wears out you are funneling water behind your siding.