Hello and welcome to the forum!
The winter season in Kinnaur is from the months of October to May. So during the month of November, you can definitely see snow. It is during the months of December to February though that the region receives the highest snowfall. Make sure that you schedule your trip at least in the first week of November before heavy snowfall occurs. Heavy snowfall can contribute to road blocks and landslides which can be quite dangerous.
The best season to visit Kinnaur is during the summer season though, from June to September. Temperatures during this time range from 9°C to 13°C which are pleasant for sightseeing. Trekking can also be done during these months as the routes and roads are safe for tourists. However, these months are also the peak season for tourists so make sure that you book well in advance prior to visiting the region.
Another great season to visit Kinnaur is the spring season, from April to May. The great thing about this season? Flowers are in bloom and give colour to the entire region. Aside from that, it's not as crowded unlike the peak season. You can get discounts in transportation and even in accommodations.
Like I've said, from December to February, expect heavy snowfall. It's best to avoid Kinnaur during these months as the roads can be very dangerous and avalanches are known to happen. Many hotels and guest houses are also closed during this time as well.
I hope this helps!