Yes, when there's a new new guy.
Common practice in most workplaces is to continue the cycle of dumping the shit work on the new guy. But, if you want to build influence, you don't do that. When the next new guy comes along, you adopt them, you show them how to get things done, and when other people dump shit on them you help clean it up. You don't do their work for them, but you also don't leave them to do the work alone.
And you teach them to do the same thing with the next new guy. When they're up to speed you start load balancing with each other intentionally. They need to leave early for personal reasons one day so you handle some of the extra workload for them. They do the same for you.
(side note - this works when you offer first - you don't open this by trying to get them to do you a favor first; you will run into people who will take advantage of this and not grasp the concept of returning the favor - when this happens you don't whine about it, you don't confront them about it, you don't even mention it, you just note that this is an indication of that person's character and you don't ever offer to help them out like that again - there will be others who understand cooperation and join in automatically, those are the people you make part of your group; so you worked extra one day and didn't get reciprocation, that's OK, it's a cheap low-risk way to learn about who that person is - scratch it off and don't catch feelings over it, be professional, you still have to work with this person you just don't have to trust them)
(second note - this does not work for anything that you might be legally liable for - if the other person has to sign their name for the work or is in some other way accountable on record you cannot cover for them and even helping may be problematic, especially in the medical field)
Maybe you're still shoveling shit, but you're not doing it alone. If you don't have a tribe, build one.
And then break the cycle.