Sherry, 90-95% (if not all) of puppies from pet stores come from pet mill breeders (sometimes called commercial breeders). Their parents are not given health care, quality food, fresh air or exercise, contact with humans unless it's for mating or taking the pups from them and are basically physically and emotionally unwell. A dam is forced to produce a litter every time she's in heat so can have 5, 6 or 10 litters before she is discarded, so the health of the puppies is like a crap shoot.... sometimes you're lucky, sometimes not.
Some pet stores actually only sell/adopt out dogs or puppies that come from shelters, so they help the issue with over population and you know the money is going to a good cause. It depends on the type of pet store you went to.
Your puppy might not be a purebred, but there is no test to know for sure. The truth is, that many mill operators aren't looking at what each breed's standard is and they mate the dogs as often as they can, with the dogs they have. They often say 'purebred' so they can take in more money for the puppy.
There are short-haired Havanese, but they don't lose their hair length as time goes on. Maybe your puppy has health issues, but not sure if you've had a blood test done for diff. things, such as liver, kidney, protein levels, etc... I'm not saying she's sick! I'm just pointing out that there could be various reasons for her coat condition. Her diet might be a factor too.
If you tell us more about your puppy, (what is her name?) we'll try to see if we can help give you answers.