The closest English equivalent I can think of is the first sound in the word "huge" if it's pronounced with a lot of exaggeration. Even that's not a perfect match.
German 'ch' in ich is a fricative, like sh in fish. or the chute in parachute.
cute starts with a plosive, a click/plop sound, like check, TikTok curfew etc
edit: I'm talking about the german "ich" meaning I, self-reference, not about how to read that word as if it was an English word, rhyming with ick(y), there you're of course correct.
