Learn English – Meaning of “their very existence”

You're not looking for the meaning of "be very much"; you need the meaning of "be very much for", which is an idiom.

To be for something is to be in favor of it, to be an advocate of it, to be a proponent of it.

To modify it with "very much" is what it sounds like: to be particularly strongly or enthusiastically in favor of whatever it is.

In this case, the teacher was surprisingly (to the narrator "I") in favor very strongly of the planned excursion.

What your advisor wants you to do is let Tom read it and ask what he thinks - if he has anything to add to your report or if he thinks the report should be submitted as it is already written.

However, does your advisor know that Tom shouldn't know about what is going on in the report? If your advisor doesn't know maybe you should have told them that.

Ultimately, what you do is up to you. Your advisor is suggesting that you should let Tom read the report and suggest changes (which, of course, you don't have to make).