Can you pay down points on an existing mortgage?
I assume you are asking if you can lower the note rate by paying points without going through a refinance. The short answer is 'no'. If you are going to apply a lump sum to a mortgage that is already in place, it should be applied to the principal. That will lower the total amount of interest you will pay over the life of the loan. Interest is calculated on the existing balance each month, so if you have a lower balance by virtue of the lump sum reduction, then the next and all subsequent payments will have more of the payment applied to the balance of the loan & less applied to interest.
If you are looking for a modification of terms because the present payment structure is beyond your means, that is a different question entirely. You may, in fact, pay some sort of modification fee to restructure the existing balance.
You might be eligible under some circumstances for a streamline refinance which is less costly than a complete refinance. Or you may be able to reduce or eliminate PMI with a lump sum reduction without refinancing the loan. A new appraisal may or may not be required.