Few suggestions I have from playing around with it:
1. It's not clear when I click the "add a budget" what it's for. There's the top-level "November 1-15" type of budget and then those budgets have categories, but the terminology could maybe use some work here. The term "budget" to me is usually synonymous with how you're using "categories" - e.g. we have a grocery budget.
2. I wasn't really expecting the ellipsis button to be a delete. I usually expect it to be like a context menu. In general it seems like the ellipsis button does different things in different locations, which is a bit of a bad pattern IMO.
3. There's a few spelling errors, e.g. "ammount" in the "add a budget" overlay
4. It's a bit weird that when you start a new budget it's immediately in the red (color-wise, at least). I feel like $0.00 left to spend should still be in the black.
5. "Track Spending" is only available for the first budget, but you can imagine that I might make next week's budget before this week is over, so it's a bit annoying that there's no "Track Spending" button on the previous budgets. I realize you can still click into the old budget, but still.
6. There's no way to edit expenses in the budget if I mistakenly add a wrong number, or if there is it's very unclear to me.
7. Negative numbers aren't allowed but you can imagine that this would be pretty useful for things like rebates or if I get a refund on something (though this is pretty related to the last item).
Anyway, love the idea for sure! Short-term budgets are pretty helpful and not something that Mint really provides, so excellent idea.