How can I "fix" a 6 inch gap behind toilet?

I'd just measure the space and cut a board to fill the gap and wedge it in between it and the wall behind the tank just below the lid

You'll need to remove the toilet tank top at some point so just put a piece of wood that is the width of the tank cover plus 6 inches right on the tank top. That way you can move it off if you need to. You can cover the wood with fabric or paint it or contact paper etc. Extra TP, tissues, non breakable flower vase with real or not flowers. etc.

What about one of those cabinet surrounds for the toilet? They make them taller too.

How about a large cloth,wicker ,wooden or plastic basket pretty cheap at craft stores or Walmart.just measure the bottom to see if it covers tank all the way to wall or at least lrg enuf not to slip thru. Or you could make something like this https://foter.com/toilet-tank-storage SOME LOW COST OPTIONShttps://www.amazon.com/s/?tag=designpower-20&field-keywords=toilet+tank+storage

believe it or not I hired an idiot that left 10 inches between wall and tank. I hate it and now am looking for a cabinet that looks decent without me spending too much. I fired the guy but since the new tile man said I would need to return toilet in order to turn it around so it would be close to wall. I can't return it because I don't have the box to the toilet. Ugh

If you or someone is handy you could wedge a wood piece behind the toilet tank and the wall. It would be sort of U shaped so that it fits around the sides and top of the toilet tank. Paint it the same color as your toilet tank. Put a hand towel or cloth over it to put your tissue box, plant or etc. and no one will know that it sticks out so far....this won't be permanent so you can remove it when you leave the rental house.

We have a space too, a guest knocked the lid off the top of the toilet into 50 pieces. Finding one to slightly fit was not easy. We put a shelf behind it so it was level with the top of the toilet and a flat basket with towels. Looks nice , eliminates the space and no more replacing the lid again!

Hire someone to fit a vanity sink and remove the normal olde one but he left a 2 inches gap between the unit and the wall. What should I do now?

Place brackets to support a shelf and install them at a height so that the shelf almost sits on the toilet back and the shelf edge is flush with the front of the tank lid. You could add a couple more shelves above this one so that the shelves look intentional, or hang a picture above the shelf. Easy to remove the shelf when necessary. Good luck!