If the stronger player is White (remove White pieces) Pieces to remove are fixed

If the stronger player is White (remove White pieces)

Pieces to remove are fixed (so openings don’t become a loophole circus).

Rating gap Δ = (WhiteStrong − BlackWeak) Low tier Mid tier High tier Strong White starts without…
Small gap Δ < 40 Δ < 40 Δ < 50 nothing
40–169 40–199 50–229 pawn f2
170–289 200–349 230–409 one knight (default g1, alternate g1/b1)
290–359 350–419 410–489 rook a1
360–429 420–499 490–579 rook a1 + pawn f2
430–499 500–579 580–669 both knights b1+g1
500–549 580–639 670–739 rook a1 + knight b1
550–589 640–679 740–789 rook a1 + knight b1 + pawn f2
590–629 680–729 790–839 queen d1
Massive gap ≥ 630 ≥ 730 ≥ 840 queen d1 + pawn f2

Why the knight choice note: Kaufman explicitly points out that giving the g1 knight is a smaller handicap than the b1 knight, so use g1 as default if you want the “lighter” version.
If the stronger player is Black (remove Black pieces)
Rating gap Δ = (BlackStrong − WhiteWeak) Low tier Mid tier High tier Strong Black starts without…
Small gap Δ < 60 Δ < 70 Δ < 90 nothing
60–159 70–189 90–219 pawn f7
160–239 190–289 220–339 pawns c7 + f7
240–329 290–389 340–449 one knight (default g8, alternate g8/b8)
330–419 390–489 450–569 rook a8
420–509 490–589 570–689 both knights b8+g8
510–559 590–649 690–749 rook a8 + knight b8
560–609 650–699 750–799 rook a8 + knight b8 + pawn f7
610–649 700–749 800–859 queen d8
Massive gap ≥ 650 ≥ 750 ≥ 860 queen d8 + pawn f7

And again: f‑pawn is the one you remove when removing one pawn.
And “pawn and move” (f7 removed for Black) is bigger than f2 pawn odds, because of the immediate tactical punishment options after 1.e4.