I'm interested in the $K$-rational points on an imaginary hyperelliptic curve, where $K$ can be assumed to be a perfect field. I don't have much experience with blow-ups in algebraic geometry, so I would like to make sure that I'm not missing any $K$-rational points, besides the obvious ones.
From the reading I've done on hyperelliptic curves, I know that projective closure of equations of the form $y^2 + h(x) = f(x)$ over $K$, where $\deg(h) \geq g$ and $f$ is monic of degree $2g+1$, have a singularity at the point at infinity, and I know that taking the blow-up of this point yields one point at infinity (unlike the real case, where it yields two points).
However, none of the sources I've consulted mention what the side-effects are of blowing up this point. If we're not adding any points at infinity, are we instead changing the unique point at infinity somehow, or adding some affine points? What do these points look like? Are they defined over $K$, and if not, what does the extension they're defined over look like?