Unable to access icloud account

I changed my icloud username and password recently as I realised I hadn't done it for a while. I am able to access my account via a web browser but am not able to sign in on my macbook. I want to sign in so that I can download photos and clear space as I have maxed out my capacity.

When I sign in it asks for my password followed by verification code. I presume this is in the password box as there is nowhere else to enter it. I have tried doing this with the most recent code I get sent but it just tells me that my password is wrong. Please could someone talk me through step by step how to sign in after updating your account? Photos of how it should look would be great as I cannot see where I should enter the verification code. I have tried everything as has my boyfriend and neither of us can work out how to do it.

Thanks in advance.