*I posted it on general poker forum but got no answer, so I post it here and see if I get luckier :)

I have a question that has come to my mind for a while without
finding a good solid answer, regarding whether to cbet or check
behind/check call with hands with A high and backdoor flush draw.

For example:

We
open to 2.5x in BTN and get called by BB. Flop is 8h 4h 3c and we hold
Ah6d. I guess we would usually check behind without the Ah, but what are
the merits here for betting or checking with the Ah? If we check behinf
with Ah6d we can comfortable call on an ace, heart, 5 and maybe 7, 8, 4
and 3, which makes a total of 29 cards if I am not wrong that we can
call on the turn.

Another example would be BTN opens, we 3bet AKo
from the BB and get called. The flop is 9h7s4h. If we hold either A or K
of hearts it's going to reduce the floats with backdoor hearts (AJ, AT,
KQ, KJ, which really are a lot of combos!), so when we bet without the
Ah/Kh, get called and no heart comes on the turn, we are in much better
shape than in the same spot with the Ah/Kh.

I see a lot of people
more inclined towards betting, but I'm not totally certain about it,
and it's a spot that comes very frequently, so I would like to hear the
opinion of wise people.

Thank you very much :)