What is the source for your information?
The first instance where the term rocket was used, in reference to the impulse engines, was in "The Cage".
The first instance where fusion is associated with the impulse engines is "The Doomsday Machine".
The SS Mariposa
had pulse fusion warp engines.
Fusion reactors are used for powering main power in spacecraft. When DS9's fusion reactor was hit in "The Visionary", the station lost main power.
So, both the warp and impulse engines power main power, and when both were damaged on the Enterprise
, the ship lost main power. Auxiliary power was damaged by other means. The question for me is this, why didn't they attempt to restore impulse power first, then warp power? Is restoring impulse power more difficult than warp power?