Feeling frustrated about work

I’m looking for some help on the following:

I have decided to limit the number of hours I work each week to 40. My team’s work is not getting done on time. I’ve asked for an additional hire, management said yes and then retracted.

I don’t like how it feels to submit work late, but also want to honor my own time. I also am debating how much time and energy I want to spend trying to make a case for additional support or discussing why the solutions they offer me won’t work, or just let management’s decision stand.  But when I think about doing this I also think, “I can’t wait for them to be wrong.”

On top of this I have thoughts about thinking and caring about work too much.