I've known it with a silent "h", so "an honest man" as opposed to "a helpful man".
The silent "h" occurs in some words of French origin:
hour
heir
honest
honour
but not in others.
whereas a hard "h" will occur in words of Germanic origin
hatchet
harness
helmet
hamlet
In terms of accents:
The French, when speaking English, will often drop the "h"s since it is not pronounced in French, e.g. Les Halles is pronounced "lay al". On the other hand, I have been asked in Paris by an American "Do you know where Les Halles (les hal-les) is? Had to think for a moment what they meant.
In BrE, an East London, South London, or Cockney accent also drops "h"s, because, well, that's what they do. So they may say "elpful" for helpful.
h
added and which wouldn't. It has been suggested that it might depend on the location origin of the word (from French of from German). As another example, consider the way that Americans don't aspirate theh
at the start of 'herb' - this was true in the British Isles a couple of hundred years ago but the pronunciation then mutated in one part of the English world and not in another. However, British English doesn't require 'an' preceding 'herb'. No reason why, but that's English for ya.