I'm not sure, as you'll need more context for this phrase.
with a high hand = arrogantly
However, "with a high hand" can also mean "with an advantage" (think of a game of cards). "High-handedly" is a slightly different meaning: without consideration of others, or an overbearing attitude.
with a heavy hand = severely, harshly. You're likely to use the verb "to deal" with these expressions (another card-game reference, of course, even if you're not deliberately using such an analogy).