You need to use the -i flag:
ssh-copy-id -i my.key.pub 10.10.1.1
From the man page:
If the -i option is given then the identity file (defaults to ~/.ssh/id_rsa.pub) is used, regardless of whether there are any keys in your ssh-agent. Otherwise, if this: ssh-add -L provides any output, it uses that in preference to the identity file