I am a far cry from being a guy who knows much about engines so please bear with me.

I have a problem with the 350 in my '72 LeMans where it will sort of make a popping noise if I hammer down on the accelerator, pause, and then take off. If I take off slowly and wait a bit before hammering down, the same thing happens.

The car is relatively new to me but what I do know is that there is a new carb on it and I suspect that could be the problem. I also believe that it has been changed to HEI and I could be wrong on that one (and I have no idea if that would have anything to do with my problem).

Advice would be greatly appreciated.

Thanks.
Otis

Make sure the wire going to the distributor is a 10 gauge wire and not the original wire of the point distributor.

Oh yes by the way, Welcome to the site

Check the mechanical advance on the HEI while you're in there. The mechanical advance mechanisms on HEI's are known to stick. Under the rotor, there are a couple paddle shaped plates that are hinged at one end and have a small spring on the other. As the distributor spins, the weights are thrown out away from the distributor shaft, extending against the spring. You should be able to move the weights (paddles) easily with a finger and they should return to be near the center shaft of the distributor when you let them go.. If they're stiff, work them a few times to loosen them up. There's a non-conducting lubricant you can apply to the pivot point.  I've used a little squirt of WD40 or better, a spray silicon lube when I didn't have the proper lube.

After checking the mechanical advance, check the timing with the vacuum line disconnected and plugged. Once it's set, I like to rev the engine a little to watch the timing advance so I know the mechanical advance is working (you shoud see the timing return to normal when it drops back down to an idle), then reconnect the vacuum and watch the spark advance so I know the vacuum advance is working.

If the timing is right, and you have a full 12 volts to the HEI but are still having problems, I'd check the vacuum to make sure you don't have a leak.  Next, I'd check setting on the power valve on the carb.  It sounds a bit like your problem could be a "lean pop" or "lean bog" which would mean the power valve is a little late squirting the fuel into the intake.   Is this a 4 bbl carb?

Yes it is a 4bbl carb and there are times when I do hear a pop.

Check the timing, but I'd look on-line for a guide to setting up your carb. Adjusting the power valve is usually pretty easy - though it's been awhile since I set up anything other than a holley. I still get a lean bog when the engine is not warmed up completely, but I figure you shouldn't be nailing it until its fully warmed up anyway...