Can a landlord kick me out if I have a child in the winter.

If you don't pay rent, or if you violate the terms of the lease in some other way, then the landlord can send you a notice to vacate and then file a lawsuit for eviction. It can happen at any time of year and it can happen whether you have children.

Of course they can, you may fight it in court and delay it 10 to 30 days, but you will be set to the curb if you don't pay rent or follow the condictions of your lease.