"Peter has a lot of room for improvement."
"The web app has a lot of room for improvement."
"Peter has much room for improvement" is correct, but less common, so will make you sound succinct and pithy if you're careful or (potentially) less fluent if you're not.
"Peter has many options for improvement" is valid, but it has a connotation that you might not expect. If you said that, it sounds like you could have also said "Peter has many options for improving," which seems to connote (facetiously, jokingly) that he's selecting one choice from a collection of possibilities, as if he were shopping or something, which, since it sounds facetious because it's uncommon, seems to say, "if he chose to improve, he could, but he doesn't." This interpretation would happen only very contextually though; it would mostly just sound unfluent.
Other related phrases include:
has room to grow,
something to work on,
a way to better (your)self, (grand connotation).
Hope it helps!