Honda has one major issue across every carbureted engine they produce.
Whether its an actual choke plate, enrichener or primer, they are too danged lean!
Their choke plates are cut so they cannot flood an engine, enricheners have too small of jetting which does not administer enough fuel to a cold engine.
Their primers are restricted as well as to how much they can squirt with each pump, or, if the primer is weak or plugged up the least bit cold starts become impossible in cold weather and the colder it gets the worse it gets.
So, what to do?? you can rebuild the carb and check the primer passages to make sure they are clear, or, carry a can of brake cleaner, ( I hate ether ) and give the engine a shot through the carb to help it out.
There are mods that can be done that requires modifying the choke/enrichener circuits but I wont type them here due to liability reasons as you are modifying a fuel system and thats a huge no no in todays world.
After all of this and its an injected model, check fuel pressures first, if its the least bit low, it will start hard especially when cold as it is running lean.