I reckon cold hands are almost worse than cold feet. When my hands get cold they hurt and as they warm back up they really hurt. If the air and water temperature isn’t too bad then I’ll go through the pain barrier as once my hands are warm they’ll stay warm. However if it is super cold then its hard to push through the pain barrier as it takes so much longer. Cold hands will limit your ability to hold stuff. If you aren’t able to hold your paddle then you are pretty stuffed.
A number of manufacturers produce both gloves and pogies that are designed to your hands warm even if they get wet.
Some paddlers like gloves but some times they can feel a little bulky and can make holding the paddle more difficult. Gloves tend to be made from neoprene and are designed to work just like a wetsuit, i.e. they work better when wet. Many years ago I got hold of a pair of Typhoon neoprene gloves that worked fantastically and still have them in my kit bag.
The other main option is to try some pogies. Made out of neoprene or nylon pogies wrap around the paddle and allow you hold it as normal. The pogies fasten onto the paddle shaft using Velcro and fit snuggly around the paddle helping to keep water out. Getting the pogies on your hands is always fairly straight forward for pogie number one but as always is a little harder when getting the second one on.
One of the main concerns people have about pogo is how easy it is to release your hands if you need to leave your boat and pull your deck strap. The opening to the pogies on all of designs I have used have never been tight, so the only thing keeping your hands in the pogies is your own grip on the paddle. If you let go you can pull your hands free.