I don't think the adjective supernumerary makes much sense for pieces that affect other pieces. The latter doesn't have anything to do with a number. In the context of chess engines I always use the term 'supernumerary pieces' to refer to the situation where you have more pieces of a certain type than were initially present, e.g. 2 Queens or 3 Knights.

You are using the word in a very literal sense, as it literally means "exceeding the standard or prescribed number." In that sense, though, supernumerary applies to the number of pieces and does not describe particular types of pieces outside of the context of a particular game.

But I borrowed the use of this word from Anthony Dickins, who uses it in a more figurative way in A Guide to Fairy Chess. After covering Leapers, Riders, and Hoppers, he has a section called "SUPERNUMERARY PIECES". He says underneath the heading:

We now come to a group of pieces that are not derived from normal chessmen but have movements or powers that are entirely new. We shall call these 'Supernumerary' pieces.

He then proceeds to describe the Imitator and Joker, which are both mimics, the Diplomat, which offers safe harbor to adjacent pieces but cannot move or be captured, and the Pyramid, which also can't be moved or captured.

He then mentions combined pieces, snipers, and hunters, which I would say are not entirely new in the same way that the first four pieces he mentioned were. This is followed by a section on special powers, which covers capturing, imitating, imitative, moving,neutrality, protean, reflecting, royalty, and invisibility. This ends his section on supernumerary pieces, as the next section has a higher level heading.

I am taking the general idea behind what Dickins meant by supernumerary but applying it more strictly to pieces that do not clearly fall into the Leaper, Rider, Hopper, or Hybrid categories. It is a catch-all for pieces with powers other than fixed powers of movement and the ability to capture by displacement.

The page doesn't really stress that how a piece moves and what it can do otherwise are really independent properties, which could be combined arbitrarily. E.g. you can have immobilizing Knights or Gold Generals. I would refer to this as 'additional powers' (as opposed to 'power of movement').

I had been considering the term "Superpowered Pieces" but felt that this term would suggest that these pieces are more powerful than other pieces, whereas "Supernumerary Pieces" would be neutral on this matter. For example, non-displacement capture usually comes at the expense of not being capable of displacement capture, and switching one for the other doesn't necessarily make a piece more powerful. I also prefer the compactness of putting an adjective before pieces over the longer "Pieces with Additional Powers". If I did go with a longer phrase, "Pieces with Special Powers" might work better, as it has fewer syllables, and the powers might be replacements for rather than additions to the usual piece powers. I'm also considering something like "Novel Pieces" or "Unconventional Pieces" for what I'm presently calling supernumerary.