None confirmed, but the best possibility is the Crup, which may be part mundane dog.
Kneazles are the only creatures explicitly confirmed as being able to interbreed with non-magical creatures. There are crossbreed creatures like Blast-Ended Skrewts, that are confirmed to be a combination of two separate creatures, but in these cases both creatures are magical.
There are also creatures that bear resemblance to mundane creatures, like a Jarvey looking like a ferret, or a Mackled Malaclaw looking like a lobster, but there's no indication that any of these creatures were created by the interbreeding of mundane creatures with magical creatures.
The magical creature that seems the most likely to be, or to have been at some point, a result of the interbreeding of mundane and magical creatures is the Crup. It's highly likely to be a creature bred purposefully by wizards, and it looks like a Jack Russell terrier.
M.O.M. Classification: XXX
The Crup originated in the southeast of England. It closely resembles a Jack Russell terrier, except for the forked tail. The Crup is almost certainly a wizard-created dog, as it is intensely loyal to wizards and ferocious towards Muggles.” - Fantastic Beasts and Where to Find Them
For wizards to have created it, that most likely means the Crup did not exist before wizard interference. How wizards created the Crup is not mentioned. Hagrid was able to create a new species by crossing a
Fire Crab and a Manticore, so it is possible for wizards to create new species by interbreeding existing creatures. It's at least a possibility, considering it was near certain to have been bred as a pet for wizards, that it was created by the interbreeding of a mundane dog and some sort of magical creatures. It's also possible, however, that the Crup was created by wizards in some other way, perhaps by breeding two different magical creatures, and its similar appearance to a Jack Russell terrier is only a coincidence, like the Jarvey's resemblance to a ferret.
The Crup is almost certainly not the product of breeding two mundane creatures. Except for the Basilisk, which is created by hatching a chicken egg under a toad which is not a typical way of being born even for a magical creature, for a creature to be magical it usually has to have some magical blood in its gene pool.
The Hippogriff of myth is the product of a mare and a griffin, but this may be different in HP.
Mythology (out-of-universe) states that the Hippogriff is the product of a mare and a griffin. This is its accepted origin in classical mythology. However, this doesn't necessarily mean that the Hippogriffs in the Harry Potter have a similar origin. It's possible that they do but also possible that they don't.
There are entries in "Fantastic Beasts and Where to Find Them" for both the griffin and the Hippogriff, and neither of them mention a Hippogriff being produced by a mare and a griffin. This doesn't necessarily disprove the idea that they might be, but it seems like something that would be important enough to mention in one of the entries, so its absence casts a bit of doubt on this being the origin of Hippogriffs in the Harry Potter universe.
However, both entries are relatively short, and may not have room to describe everything about the creatures they're about. In addition, the descriptions of both the griffin and the Hippogriff are similar to their mythical counterparts, which makes the idea that their origins might be similar as well more plausible.