You might be able to use water as a reactant to combine with the vessel's fuel (in effect giving you bonus fuel because you get reaction mass from the outside), but you are wanting to use water as the only fuel source.
Unfortunately, water is fairly stable. It is a product of the decay of energy-rich compounds. It is less energetic than elemental hydrogen and oxygen gas (which is why, when you burn hydrogen, energy and water are released).
This means you can't use any chemical process using only water for fuel to generate more energy than you put into the system. It doesn't matter how you try to break the water apart, the laws of thermodynamics require you to pay for lunch. There ain't no such thing as a free lunch.
The vessel would need to go nuclear
There is, in fact, a way that to extract more energy from pure water than you have to pump into it. Fusion. The first problem is we don't have working fusion reactors, so this requires sci-fi future tech.
The next problem is you want a jet engine. Even if you have futuristic fusion plant technology, this would be a plasma jet engine, and the problem with a jet of radioactive plasma exhaust hot enough to undergo nuclear fusion is it will be essentially impossible to contain in an engine. The plasma will destroy the engine.