To answer your question: whether the second mortgage can take the surplus and keep their lawsuit going depends entirely on whether they are a "superior" or "subordinate" lien compared to the HOA.

If the 2nd Mortgage is Superior (Recorded BEFORE the HOA lien): The HOA auction did not wipe them out. Their foreclosure lawsuit is still completely valid, and their lien is still attached to the property you bought. However, under Florida law, superior lienholders generally cannot claim surplus funds from a subordinate auction. If they do, it just lowers the payoff amount, but they can still foreclose for the rest.

If the 2nd Mortgage is Subordinate (Recorded AFTER the HOA lien): The HOA auction wiped out their lien on the physical property. Their foreclosure lawsuit against the house must be dropped. Because they lost the house as collateral, they are now legally first in line to claim those surplus funds to satisfy the old owner's debt.

Warning: Do not pay off the first mortgage yet! If the second mortgage survived the auction (is superior), paying off the first mortgage will simply bump the second mortgage into the first-place position, and they can still foreclose on you. Get a title search from an attorney immediately to confirm the exact recording dates of all liens before spending any more money.