Two small points; the Merlin-powered versions were the P-40F and P-40L; the P-40K was the follow-on to the P-40E, still with the Allison engine. One easy way to tell if a P-40 has a Merlin or Allison is to look at the upper cowling; if there's an air-intake it's an Allison,
if there's no air intake, it's a Merlin.
The British designations for their versions of the Curtiss 81/87 series (equivalent to the US P-40) were fairly confusing, and don't correspond strictly to which engine was installed. For example, the Kittyhawk Mk.III could have either a Merlin or Allison. Curtiss changed the model number of the P-40 series from Curtiss Model 81 to the Curtiss Model 87 with the P-40D, and that was where the change from Tomahawk to Kittyhawk occurred. The Tomahawks were Model 81s; the Kittyhawks were Model 87s. Pete Bowers' Curtiss Aircraft 1907-1947 gives the following breakdowns for British variants:
Tomahawk Mk.I: equivalent(except for British equipment, guns, etc.) to the P-40
Tomahawk Mk.IIA: equivalent to the P-40B
Tomahawk Mk.IIB: equivalent to the P-40C (100 of the 930 earmarked for the RAF were released to China for the AVG)
Kittyhawk Mk.I: equivalent to the P-40D
Kittyhawk Mk.IA: equivalent to the P-40E
Kittyhawk Mk.II: equivalent to the P-40F or P-40L, depending on which batch they were from
Kittyhawk Mk.III: Here's where it really gets complicated; equivalent to the P-40K-1 (first 192), P-40L (next 160), or P-40M (last 264).
Kittyhawk Mk.IV: equivalent to the P-40N.