Provia 400X exposed accidentally as ISO 500 -- to push or not to push?

So I decided that I'd rate my Provia 400X as ISO 800 and then push process it one stop. Unfortunately I had shot some Portra 400 in my F100 before this roll and so I had +0.7 exposure adjustment set (while ISO was 800). My question is: should I process it normally (and have it 1/3rd stop underexposed) or should I push process one stop? How much does push processing affect in this kind of situation?

Are you sending it to a lab? There are labs that will push/pull in half stop increments. Of course, they charge you a bit more. If it were me, I'd just process it normally.

It depends on your subject matter and your "real" exposure. If it benefits from being a bit under exposed, process as is. If you need the extra shadow detail, push 1/2 stop. Sadly, many labs are not THAT exact, mind you, since they use replenished tanks.

A 1/2 stop push will give you a hair more grain, and your blacks may turn a shade of brownish black in the worst case, depending on their chemistry balance and process for "pushing."

As said above, process "as is" is usually fine with larger labs, which tend to be softish anyway.