Daily entries from the 17th century London diary
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“A game at cards not unlike Loo, but with this difference, the winner of one trick has to put in a double stake, the winner of two tricks a triple stake, and so on. Thus, if six persons are playing, and the general stake is 1s., suppose A gains the three tricks, he gains 6s., and has to ‘hand i’ the cap,’ or pool, 4s. for the next deal. Suppose A gains two tricks and B one, then A gains 4s. and B 2s., and A has to stake 3s. and B 2s. for the next deal.” —Hindley’s Tavern Anecdotes. — M. B.
http://www.davidparlett.co.uk/histocs/loo.html
This excellent link explains the rules of loo, an historic card game of which ‘Handycapp’ is a variety.
Hence, if you win one hand, you are “handicapped” by having to put in a bigger stake for the next. The SOED has a different explanation of the game and origin of the phrase, but at about the same date.