Cursor (round)
The round counter that steps to a fresh HMAC output once a game has used up the bytes of a single 32-byte hash.
Each HMAC-SHA256 call yields 32 bytes. At 4 bytes per float, that is enough for 8 floats. Dice and limbo consume a single float, but games like plinko on 16 rows, mines with a high mine count, or card games consume more.
Mechanics
The usual message format carries a round number: clientSeed:nonce:round. Bytes are read from round 0 first. Once those 32 bytes are exhausted, the generator runs HMAC again with round 1, then round 2, and continues as needed. The cursor is simply the pointer to where you are in that byte stream. Depending on the casino's docs it may be labelled cursor, round or current round.
Effect on verification
Try to check a 16-row plinko drop with a single HMAC output and only the first 8 rows will match; the remainder will be wrong. A decent verifier steps through the cursor for you.
Example
Suppose a mines bet consumes 24 floats, which is 96 bytes. The verifier has to compute rounds 0, 1 and 2, at 32 bytes apiece, and read floats sequentially across all three.