No, this is not correct. The closest correct sentence you can make from this is:
He incorrectly counted more money than I actually gave him.
"Wrongly" and "incorrectly" are often synonymous, but not always, and not in this case.
- "He incorrectly counted" means that he made a mistake in his count.
- "He wrongly counted" means that he made a mistake by counting the money. The counting itself was the mistake. He shouldn't have counted it.
That being said, because of the continuation of the sentence, most people will understand what you mean if you had used "wrongly".
I removed "the amount" because it's unnecessary and a bit stilted, money is inherently understood to express an amount, especially if you're already quantifying the amount of money use by using "more".
"Actually" doesn't have to be in the sentence, but it is idiomatic here. It's used to further stress that there is a difference between the real amount and the mistaken amount.