The suffixes -kinesis and -kinetic seem to have been naively derived from the well-known "telekinesis"; they actually mean "motion". Literally, "Hydrokinesis" would be the power to move water.
Similarly, -mancer and -mancy seem to have been derived from the well-known "necromancy", even though in the original Greek they come from the word manteia
, meaning "divination". The "-mancy" template originally referred to the practice of divining the future using whatever the prefix was as a medium. So a classical necromancer would ask important questions of a dead person's spirit or read the entrails of a sacrificed (dead) animal, a pyromancer would look for important symbols in flames, and so on. The proper suffix for someone magically manipulating a substance, force, or entity would be "-urgy" and "-urge" which derives from érgon or "work".
This, of course, does not change the fact that in practice, "-mancy" and "-kinesis" are ubiquitous and "-urgy" is nearly unheard of, aside from the occasional "thaumaturgy" as a fancier word for "magic".