The video shows that the play seems to be in the hub:
Is it the bearings causing this? Are they worn out or not adjusted?
What steps can I take to resolve this play?
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Yes, this is play in the bearings. How to fix it depends on the type of hub you have.
'Cup and cone' bearing hubs have ball bearings between fixed cups in the hub and adjustable cones on a threaded axle. These can be adjusted to take out play.
Other hubs have pressed-in cartridge bearings. These just need to get replaced , you'll probably need a bike repair shop to do this for you as they will have the required tools.
You may be able to do some research and find what type of bearings you have, otherwise you'll need to start disassembling.