We want to travel in December to NYC for 1 to 4 nights. With a child it would be very helpful to have some sort of kitchen set-up (just a microwave and fridge is enough) and ideally a suite with a separate room, though a room with 2 beds and a bit of extra space -- if it had the kitchenette -- could do.

Does anyone know of anything like that? Ideally under $350 per night.... though we are braced for up to $400.

Hundreds of hotels in NYC fit that description. Off the top of my head:

The Affina Hotels (Shelburne, Surrey, Dumont, etc)
Carnegie Suites
Salisbury Hotel
Doubletree
Embassy Suites

Lots of options but few at your price range. You may have trouble finding a hotel room for $400 this month. The cheapest time tends to be start New Year's until the end of the year. Before that, I doubt you'll find many options. I've been surprised at how busy the city is this year.

If you can wait til later in the month (after the 17th) you can find what you want. ALL of the Affinia hotels are under $350/night from 12/17 for a one bedroom suite. Studios (one large room) are even less.

The Blakely is another wonderful hotel with a 1 bedroom suite for $325/night, again starting the 17th.

Should have mentioned those prices are on affinia.com for the Affinia hotels and blakelyhotelny.com

Quikbook.com sometimes has even better prices so worth checking. They have the Salisbury for $240/night for a 1 bedroom suite which is really great value but with quite large rooms. It's not as nice as the others but in a great location near Central Park.

We just stayed at The Kimberly (E 50th btw Lex & 3rd). It had a one bedroom suite with kitchen that was nice, as well as a standard room with a microwave and fridge. However, the gas was off at the hotel and we were not able to use the stove. I think the rate falls in your budget.

We loved the Blakely this past spring for our family. It is a small hotel with a wonderful staff who learned all of our names within minutes. Be sure to request a bedroom with two queens. (There is a separate sitting room with kitchenette and large updated bathroom.)It's in a great location and near lots of neighborhood restaurants which deliver. Just perfect for kids!