Personally, I think your teacher is wrong. But that's just my opinion. To me they're exactly the same, but still, neither is really clear. At least, I don't know what free snacks and bad bread have in common here. But I think your first sentence actually sounds a little more natural here, although both expressions are OK. Maybe "didn't like how Mom allowed..." would sound slightly better in day-to-day speech. I suppose in formal writing "Mom's allowing" like you wrote is better, but it sounds stilted to me. "Mom allowing" would be better in casual speech or writing in my opinion. Anway, here's a somewhat random source for that: "Possessive with gerunds" https://www.btb.termiumplus.gc.ca/tpv2guides/guides/wrtps/index-eng.html?lang=eng&lettr=indx_catlog_p&page=90SDm90b7Xlw.html

(edited to remove answer that was wrong :o ) Note: "bad bread" isn't a phrase I've heard before. The store owner could have been selling stale bread, spoiled bread, bread that tastes bad, bread that was badly baked, bad quality bread, or maybe bread that Mom just really very strongly disliked.

I think both are grammatically correct. but I agree with your teacher that "that" can be removed so the sentence flows nicely.

Hi Ppman, I would not recommend to use the word “allow” in this context. You don’t allow someone to get away with something but you let them get away.