Do I have to move the mulch before adding compost and fertilizer?
If you use a mulch that has the correct texture and has been composted first, this should not be a problem. I simply “topdress” (spread compost and all-natural organic fertilizer right on top of last year’s partly decomposed mulch) then replenish the mulch as needed.
The problem with most mulches, even bagged shredded bark, is that they haven’t been composted before they are sold and/or are too coarse, so they don’t break down very well, but rather form a sort of coating on top of the soil.
Good mulch, on the other hand, breaks down and improves the soil below. I don’t mean it breaks down in a week or a month, but over the course of a season or two.
If you have a thick layer of mulch that’s not breaking down nicely, and the layer is just getting thicker and thicker each year, rake some away and then add your amendments and start using a new, better-quality mulch this year on top of that.