Poplar helicopter seeds stuck in deck gaps

You'll need an engineer to tell you if those posts have enough good material remaining. But one thing you can do is figure out if your post sizes and spacing is over-built for your local codes. You'll have to find the codes for your area.

Here is an example of deck specifications.

If they are OK, you'll want to re-grade and treat them to prevent more rot.

If they are not OK or you just want them to look better, you can cut off the bad portion and pour new (higher) footings. (Make sure the footing depth and diameter and height above grade meet local code) Make sure you cast a metal post base in the footing. Don't cast the wooden post in the footing.

I have installed membranes underneath decks and ran them to gutters to provide an under deck patio space. To answer your question, no you can't do that - for the easiest reason - because it won't work and your guests will get wet. And for a hard reason, it will tear up your deck with wood expanding and warping due to pressure or wood rotting due to standing water.

I would ask a follow up question that asks possible ways to divert water, so that people can answer that.