Trump increased his overall numbers both in NYC and most of its surrounding suburbs.
The source is the actual vote count and not an exit poll and thus can't be broken down based on gender.
I believe Trump did better in the cities and the suburbs compared to 2016, with the exception of the Milwaukee, Philly, Atlanta, Detroit, Las Vegas and Maricopa metro areas. For some unexplained reason (ha!) those metro areas bucked the trend.
Trump did better percentagewise in most cities the second time - cities outside of the swing states. Not enough to matter though.
Not so much in the suburbs. There was a falling off of his vote in most suburban counties. Many suburban voters wanted things to return to “normal.”
Trump did better in some NY suburbs in 2016 than Romney did in 2012, but it looks like he fell back in 2020. Whether that was legitimate or otherwise, I can’t say for sure, but probably the same dynamic was going on there as in other suburbs across the country.