I could not find a "Henry Barham" on the microfilm for the 4th USCHA, but I did see that they were filmed somewhat out of alphabetical order. (evidently a researcher handled and disarranged the box before filming). The only 'Sharp' that I saw in the 4th was David.

There were only 2 Barhams in the microfilm index to USCT (M589), Henderson Barham in the 59th infantry, and Miles in the 31st infantry.

I didn't have time to check for ant alternate spellings (Barton, etc.)

Not having an accurate regiment and being unable to prove her husband's service might be the reason that Mollie's pension claim went unapproved.