Would it explode in a supernova or will it form a new star?
Fun question. Without doubt it would be very violent and spectacular, but not much is known about stellar collisions and only a few have ever been observed.
Most stellar collisions happen due to tight orbits where the stars spiral in towards each other or perhaps, 3 or more body chaotic star systems with unstable orbits that lead to an impact. Space is so large that actual star on star collisions are exceptionally rare.
A spiral in towards each other is almost certainly more common, it also quite different in terms of outcome. Much of the gravitational kinetic energy of the collision goes into angular momentum, so, while still spectacular, it's quite a bit less explosive than a direct head-on collision.
Black hole/black hole collisions have been observed by detection of gravity waves when two black holes orbit around each other and the orbit decays.
Neutron Star/Neutron star collisions may be responsible for the very heavy elements in the Universe and White dwarf-White dwarf collisions may lead to a supernova similar to a type 1a Supernova (per Wikipedia source). All these collisions.
In a tight orbit, a white dwarf will steal material from a near-by star and if the orbit is close enough, lead to a decaying orbit and collision, but that would happen over time. If the white dwarf reaches the Chandrasekhar limit in that scenario, it will undergo type 1a supernova. If there's not enough mass for that, it should eventually absorb the nearby star, leading to a new star, admittedly a strange one with very high metallicity and perhaps abnormally hot and short lived for it's mass, but it should basically be a star once the collision settles down, but probably a star that burns both hydrogen and helium, but wouldn't collect enough helium for a helium flash. I don't see any other alternative with that kind of below Chandrasekhar mass spirally in collision than it being a very metallic star, hot for it's mass and with a shorter lifespan.
Now a head on collision - that's very different and, I'm basically guessing, but you have a white dwarf star - about the size of Earth with a mass probably at least 30%-40% of the sun. White dwarfs don't have a specific minimum size, but for a star to go white dwarf within the 13.6 billion years or so that stars have been around, that star would have needed enough mass to have completed lifecycle and in the process, it loses some mass, so perhaps, 30% of the mass of the sun might be the current lowest possible mass of a white dwarf, but as billions of years pass, that low end will grow smaller.
But if we take a low-mass white dwarf at 30%-40% the mass of the sun, it's still a super-dense object and crashing that into a star at their combined escape velocity, not the more common decaying orbital spiral, that's a messy calculation. I'd only be guessing.
Escape velocities aren't additive, but still, white dwarf and small star would collide in this theoretical at an impressive velocity. If it was a glancing blow, the white dwarf would almost certainly punch straight through, blowing a lot of matter off the star and perhaps affecting it's rotation rate. It would be an impressive site.
If it was a direct hit, I don't know if the white dwarf would punch through the star or not. It might still punch through. The White dwarf would pull the surface of the star towards it as it got close and that would be the first contact between the two. The Sun's escape velocity is about 618 km/s and the mutual gravitation the White dwarf would be traveling to the sun a bit above that, but the White dwarf's surface gravity is much greater, so the mater pulled off the sun into the white dwarf would be hitting the white dwarf at a few thousands of km/s. (Small white dwarf, so lets say 3,000 km/s).
That's still far too slow for fusion to happen, so this is basically still just collision mathematics. As the white dwarf plowed deeper into the star, towards the much hotter core, the heat and velocity of impact probably would increase fusion rates, but if it just passed right through the star that would be a pretty temporary and minimal effect. The larger effect would be the disruption to the star after a very dense planet sized object punched it's way through it.
If the star was viscus enough to stop the white dwarf from making it all the way through, then the combined mass and dense center and added heat from impact should significantly increase the rate of fusion of the original star, even though some material would likely be blown off, the net result should be a heavier star that burns hotter and faster and if the new mass is comfortably above 1.4 solar masses, you likely have a super-nova in the making. But as to whether or not a white dwarf would be to punch all the way through a star upon direct collision - I have no idea. I guess that it would, but that's just a guess.
A fun related question is, how long would it take the white dwarf to emerge from the other side, assuming it's a relatively center hit and assuming it passes through (which it might, but I'm not sure if it does). I'll try to give that a look too.
Anyway, just my 2 cents on this question. Corrections welcome.
I'm far from an expert in this, but I have a reasonable guess at an answer. The White Dwarf page on Wikipedia describes what would probably happen. It seems that the WD would consume mass from the companion star until it can either sustain fusion, or fuse so much so quickly that it explodes into a Type 1a supernova.
If a white dwarf is in a binary star system and is accreting matter from its companion, a variety of phenomena may occur, including novae and Type Ia supernovae. It may also be a super-soft x-ray source if it is able to take material from its companion fast enough to sustain fusion on its surface. A close binary system of two white dwarfs can radiate energy in the form of gravitational waves, causing their mutual orbit to steadily shrink until the stars merge.
It does appear to depend on the mass of the companion, and whether they actually collide or merely pass close. If they didn't form together, it is unlikely that they will be captured in each other's orbit later on, so the WD might just pass through the system, disrupting planets as it goes but leaving the other star unharmed. There doesn't appear to be a one-size-fits-all answer here.