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We know that Crookshanks was a hybrid, a kind of half-magical animal (half cat, half Kneazle). Did any Harry Potter sources ever mention the existence of other such half-magical animals or breeds?
Were they frequent? More specifically, as frequent among animals as half-blood magic users were among humans?
I am guessing Hagrid doesn't count since giants aren't "creatures", same with Veela.
Well, there is the centaur, which is half human and half horse, and has magical abilities:
The ways of the centaur are shrouded in mystery. They are generally speaking as mistrustful of wizards as they are of Muggles and indeed seem to make little differentiation between us. They five in herds ranging in size from ten to fifty members. They are reputed to be well-versed in magical healing, divination, archery, and astronomy. (The latter two may not technically be considered "magical"¹)
Fantastic Beasts and Where to Find Them - Newt Scamander - page 6 - Scholastic
It would seem that the centaur might not qualify, but the centaur is classified as a "beast" and not a "being" by the centaurs' own request.
The centaurs objected to some of the creatures with whom they were asked to share “being” status, such as hags and vampires, and declared that they would manage their own affairs separately from wizards. A year later the merpeople made the same request. The Ministry of Magic accepted their demands reluctantly. Although a Centaur Liaison Office exists in the Beast Division of the Department for the Regulation and Control of Magical Creatures, no centaur has ever used it. Indeed, “being sent to the Centaur Office” has become an in-joke at the Department and means that the person in question is shortly to be fired.
Fantastic Beast and Where to Find Them - Newt Scamander - page xiii - Scholastic
There's the Chimaera, which has a lion's head, a goat's body, and a dragon's tail. Trouble is, Fantastic Beasts doesn't tell us what's magical about it, except to note that it's very dangerous.
The Manticore has the head of a man, the body of a lion, and the tail of a scorpion, and is said to croon as it eats its prey. It's hide repels all charms.
The sphinx has a human head on a lion's body and is skilled at riddles and puzzles. It becomes dangerous when the treasure it's guarding becomes threatened.
I'm not sure if this is what you had in mind, but these are the examples of two or more species bred together that have magical powers.
ETA: Pursuant to your clarification in the comments, my answer is now No, there are no other known half-Muggle, half-magical animals or creatures that I can find. If I come across anything, I'll edit it into this answer. But for now? No. Nothing.
¹My note
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.
“CRUP
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.
In addition to Slytherincess magnificent answer (as usual) I would like to add the not stated as a half-breed but indeed have muggle parentage. The all so famous basilisk and it doesn't say anything of magical parents at all. So in fact this is a magical creature born from two "muggle" creatures.
Of the many fearsome beasts and monsters that roam our land, there is none more curious or more deadly than the Basilisk, known also as the King of Serpents. This snake, which may reach gigantic size, and live many hundreds of years, is born from a chicken's egg, hatched beneath a toad. Its methods of killing are most wondrous, for aside from its deadly and venomous fangs, the Basilisk has a murderous stare, and all who are fixed with the beam of its eye shall suffer instant death. Spiders flee before the Basilisk, for it is their mortal enemy, and the Basilisk flees only from the crowing of the rooster, which is fatal to it.
Most macabre beasts
Which was written on the note Hermione tore from the book in Chamber of secrets.
