Is there an engine which can give a breakdown of its evaluations Engines evaluate a position based on a number of factors - material imbalance, king safety, control of the center, pawn structure etc.
My understanding is that they then combine these separate evaluations using a weighting for each (which can be adjusted), to produce a final score for the position.

Is there an engine which can display a breakdown of the separate evaluations?

For example:

Material: 0.00
King safety: -1.21
Center control: +0.87

Which means that material is equal, but black's king is less safe than whites, and white has better control of the center.
Yes. Look at Stockfish 6. You can try it yourself.

EDITED

MG -> Middle Game
EG -> End Game

Most chess engines have a different set of evaluation or algorithm for an endgame. They could use it to model phase transition (e.g.: middle-game to end-game).