The weather in January - February will not allow you to sunbathe and swim in the sea and open pools, but favorable sea air, lower cost of hotels and apartments during this period attracts tourists to Cyprus at this time of year.

When choosing accommodation, you need to decide what is more important for you - the hotel infrastructure and indoor heated pool or economical accommodation (there are no private heated pools in the apartments and private villas.)

As for the region: in winter, 70% of hotels and apartments and, consequently, infrastructure facilities are closed in Ayia Napa, so I would not be attached to this region in winter. In addition, you are unlikely to swim in the sea, and therefore the advantage of Ayia Napa on the beach here will not be significant.

I recommend to consider hotels in any of the regions of Cyprus, focusing on those that provide a discount when placing an adult and a child .The second child as a baby up to 2 years will be staying at the hotel and so it's free .If it is important for you to have a kitchen in your room, then you need to take apartments in a complex of good standard, in order to avoid the presence of insects in the room .Although in Cyprus this is not a disaster, but the apartment is not excluded their presence, as they are on the 1st or 2nd floors and with poor cleaning and the availability of food crumbs on the floor, the ants can appear in the room .