more often than not, many things in the business are on fire and underfunded at the same time. you can get recognition for your work without the problem being permanently solved the right way, and it may not result in more funding but peopel will think of you for new opportunities that pop up later as someone who is reliable.
if you dont think the recognition will happen and youre just burning out solving these problems then stop solving them. new problem pops up thats outside your job description, its not your problem. generally though if youre working for someone like that anything you do is a lose-lose
Repeatedly requesting time/budget to fix an ongoing issue is a requirement of any half-decent manager. If they’re reporting issues then just smiling blankly when asked “what can we do about it?” they’ve failed their basic job duty.