Mine are parakeets, but I had at least a couple of months before I could let them out so I could clean the cage, so I know how awkward that can get. And, today, I went in to tidy up while both of them were out in their enclosure. Mustrom actually flew directly back into the cage (normally he stops on the front porch and then hops in) and sat on the center perch, watching me while I cleaned one half of the cage. Brave soul that he is, he left Esmerelda to fend for herself outside the cage.
But she prefers to be out of the cage, and would rather watch from the top of the cage anyway.
So much of this will depend on your cage set up (e.g. I have paper on top of the grate, so I have to pull that out which involves a lot of rattling and scrunching the paper up, rather than just sliding a grate out and giving that a scrub), but I think the principles are generally applicable.
First, I don't always say, "housekeeping, come to tidy up" cause that'd be silly, but I try to be in that mindset of being there to do a necessary thing and do it as unobtrusively as possible. Same way when I reach in to change out their food or water every day. I don't try to be sneaky, but I do aim for quietly competent.
Second, I pay attention to where they are. I make sure I'm not blocking their line of retreat if they want to move to another part of the cage. I also find that mine are more okay with me doing things under them than reaching over them. So, if they're on the floor or a low perch I work on another section or wait and come back later.
To the extent reasonable, if a perch needs to be cleaned, I try to take it out and clean it outside the cage (I have a bunch of extra dowel perches, so I just usually swap those in as I go). Also, If a perch or a feeding dish is likely to make it harder to do something (I have perches almost at floor level) I usually pull those out so when I'm pulling up the paper it doesn't get caught on the perches.
If something needs to be cleaned inside the cage, I get a cloth or sponge wet and put the soap or (bird safe) cleaning solution on the cloth rather than spraying in the cage and then clean with that and wipe up with a new wet sponge or cloth.
I use large sheets of paper on the floor. It's awkward for me to reach in and get the corners of those so that everything doesn't just spill onto the next sheet. My birds don't like the crinkling and the large paper monster rising up from the floor. So, I usually also put smaller papers under the spaces where they usually poop or drop food or whatever. That way I can spot clean those sections by just taking up and replacing the smaller papers without having to wrestle with the big sheets every day. The only thing worse than taking an individual sheet out is putting a big sheet down, so I put a stack of them down at once and then just pull up the top sheet as needed.
That's all that I can think of.