I suspect that, given the choice of an extra piece or an extra couple of pawns, a chess master would usually prefer the piece - unless those were the only things left on the board besides the Kings, in which case the promotion power of the pawns would give them an edge.
As Philidor1792 shows in the following game, at the club level the pawns can be for choice in the tug-of-war with a piece. We have seen many times previously that Philidor1792 is not afraid to exchange Queens in the Jerome Gambit; and here he gives a good example of how to press ahead.
Just follow along.
Philidor1792 - mihailsoloha,
Casual Game, Chess-samara.ru, 2014
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+
4...Kxf7 5.Nxe5+ Nxe5 6.Qh5+ Kf8
7.Qxe5 Qe7 8.Qxe7+ Nxe7
9.c3 Bb6 10.d4 d6 11.Nd2 Bd7 12.Nc4 Bb5 13.Nxb6 axb6
14.Be3 Kf7 15.Kd2 Rhf8 16.f3 Kg8
17.b3 Ng6 18.c4 Bd7 19.Rhf1 Nf4 20.g3 Ng6 21.Rf2 Rf7 22.Kd3 Raf8 23.f4 Ne7
24.f5 g6 25.g4 gxf5 26.gxf5 Kh8 27.Rg1 Rg8 28.Rxg8+ Nxg8 29.Bg5 Nf6 30.h3 Nh5 31.Ke3 Kg7 32.Rg2 Kf8
33.Bd8 Be8 34.Bg5 Nf6 35.e5 dxe5 36.dxe5 Ng8 37.Kf4 h6 38.Bh4 Rg7 39.Rd2 Rd7 40.Rd5 c6 41.Rxd7 Bxd7
42.e6 Be8 43.Ke5 Kg7 44.f6+ Nxf6 45.Bxf6+ Kg6
46.Bd8 b5 47.cxb5 cxb5 48.Kd6 Kf5 49.Bh4 Kg6 50.Ke7 Bc6 51.Kf8 Kh5 52.Bd8 Kg6 53.e7 Kf5 54.e8Q Bxe8 55.Kxe8 Ke4
56.Ba5 Kf3 57.h4 Kg4 58.Be1 Kf4 59.Kd7 Ke3 60.Kc7 Ke2 61.Bb4 Kf3 62.Kxb7 Kg4
63.Be7 Kf5 64.a4 bxa4 65.bxa4 Ke6 66.Bd8 Kd7 67.Bb6 Kd6 68.a5 Kd5 69.a6 Ke4 70.a7 Kf5 71.Be3 h5 72.Bg5 Kg4 73.a8Q Kf3 74.Qa4 Kg3 75.Qf4+ Kg2 76.Qe3 Kf1 77.Qd2 Kg1 78.Kc6 Kf1 79.Kd5 Kg1 80.Ke4 Kf1 81.Kf3 Black resigned
