an earnest plea for favor: a request that rests entirely on the kindness of the one addressed, made by someone with no claim of their own to stand on
This word almost always appears in the plural, and it does not mean prayer in general. It names one particular kind of plea: the appeal that rests entirely on the kindness of the one being asked. It is built from the same root as the Hebrew words for undeserved favor and for the God who is gracious, so the request has its own answer folded inside it. To make this kind of plea is, by definition, to ask for favor rather than to claim a right.
This is why the word gathers around moments when the one praying has nothing to bargain with. Daniel states the logic outright: the people bring their pleas before God "not because we are righteous, but because of your great mercy" (Daniel 9:18). In the Psalms it settles into a near-fixed phrase, "the voice of my pleas for mercy," the cry of someone waiting to be heard (Psalm 28:2; Psalm 130:2).
The same shape appears on the human level in the word's two non-prayer uses. A poor man "speaks pleadingly" because poverty leaves him no leverage (Proverbs 18:23), and God asks whether the sea-monster will ever beg for mercy the way a captive would (Job 41:3). In both, the plea belongs to the one who has no power but the goodwill of another.
It sits close to the ordinary word for prayer (תפלה) and to its feminine near-twin (תחנה), but it is narrower than either: not prayer as such, but the specific plea that throws itself on mercy alone.