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I was thinking whether you could move the sun by an engine that sends a a lot of energy and matter in a direction which will result in the sun to move. The material would be coming from the sun.
This requires taking hydrogen and helium and sending some of this material into the sun to prevent the device from crashing in the sun. Then the other material will be fused together to produce a lot of energy and be what propels the sun.
What is the reality of this idea.
The most obvious stellar engine is the Shkadov thruster.
Essentially: solar wind has an enormous amount of momentum if you’ve got a big enough parachute. By building a vast (and I do mean vast) mirror across part of your solar system you can balance the inward force of the star’s gravity with the outward force of it’s radiation. Your mirror ‘hovers’ in place.
This in turn creates a radiation imbalance. Your star is throwing more stuff in one direction than another. Over sufficient timescales this will start pushing your whole solar system in the direction of the mirror.
When it comes to pulling fuel from the star: Don’t. Stars are more efficient engines than you could reasonably hope to build, it’s just that they’re thrusting in every direction at once. Find a way to direct that thrust instead.
The easiest option would probably be a Shkadov thruster built using a light-sail gravity tractor approach. You build a half-sphere of mirrors around the sun at a distance where light pressure exactly counteracts the gravitational attraction of the sun. Since the photons escaping that system all move in the same general direction (they still cover half a sphere, but their movement along the 'x-axis' (depending on where you put it) always has the same sign), it consistently accelerates sun/light-sail system. This design has a significantly lower efficiency than if one reflected/bent all the light to go into one coherent direction, but wouldn't need any significant active stabilization (if your mirrors are not in the way of a planet).
You could get way more efficiency out of a similar design if you aligned the mirrors in a way so that all photons would be deflected in a single preferred direction (think of a faceted or parabolic mirror, and perhaps flat deflector arrays to even align the other half of the total stellar radiation), but since the mirror would now be accelerated away from a direction halfway between the orbital radius and the direction of the beam, combined with gravitation it would experience a net force trying to crush your mirror together (towards the axis of the emitted light). Compensating for that doesn't seem trivial.
To get the most out of a stars energy production, you'd do well not to rely on photonic propulsion. Since photons have no rest mass, they are incredibly energy inefficient per unit of momentum transferred. And since you are moving a whole star, you probably have more mass than you actually need. Ideally, you would gather heavy elements from the core of the star and accelerate them magnetically using energy gathered by a "common" dyson sphere (or dyson swarm), but it would be way easier to gather the material to be expelled from the surface of the star. You'd get mostly hydrogen and helium this way, which you should fuse into something heavier (ideally iron) to prevent wasting potentially usable energy. The Caplan Thruster design (bussard ramjet fusor engine fed by induced solar wind) might be a quite efficient example of this kind of propulsion (in terms of complexity compared to fuel efficiency), though achieving fusion (and capturing/directing the energy released by it) to something closer to iron (the most stable element) would get you more momentum per used mass.
Depending on how exactly black hole evaporation works, the most efficient possible thruster might use a Kugelblitz black hole being fed siphoned off stellar matter. If its emitted radiation could be deflected roughly equally towards the star and away from it, it would remain stationary while photonically pushing on the star. Capturing the released energy and accelerating massive particles would not be any more efficient in this case, since you'd already be emitting half of the total mass energy of the matter into space (the other half being required to push on the star).
Using the black hole installation as a gravity tractor might be even more efficient, possibly emitting 100% of the mass energy of the material consumed in one coherent direction. It would have to be aimed around the star, requiring additional mirrors or incurring a slight (<1%) performance penalty, but this approach would get you as close to perfectly energy efficient propulsion as possible under our current understanding of physics.
I should probably not that while according to our knowledge black hole propulsion might very well be possible, current physics also predicts some specific challanges (which are far less relevant when moving massive stars, but should still be considered), including the emission of extremely highly energetic photons (or even massive particles) by smaller, higher-output black holes and possible problems feeding a black hole that is radiating many TW of power while having a Schwarzschild radius in the attometers.
