meself,

A capstan has several small pawls at its base, around its circumference, to prevent it from revolving backwards. As the sailors heave (push) the capstan around in the forward direction, the pawls are continuously lifted out of their notches and then fall back into the next notch that comes around. The constant "ping ping" sound of iron against iron in this video is the sound of the pawls falling into the notches.

A brake windlass has one great pawl. It's the big rectangular piece of iron that is falling into the notches at the start of this video. .

"Heave a pawl" literally means to heave (push) the device so a pawl's-worth increment of rotation occurs. Of course, in practice you'll be rotating the device much more than one pawl's worth, so the phrase is more figurative in the songs and more just like "Heave!"(with the possible implication that the anchor's stuck, work has slowed to a crawl, and you've got to put an enormous effort just to gain an additional pawl's worth of progress).

"Heave AND pawl" is something different: An order, at the completion of the capstan maneuver, to push the capstan forward just enough to be sure all the pawls have fallen into their notches (and thus secured) rather than in halfway-up position. Otherwise, the capstan might lurch backwards if you suddenly ceased all pressure in the forward direction.