I am not sure what
list_of_my_users_obtained_earlier looks like, but assuming it is just a normal list of just usernames, you probably want something like:
authorized_key: user="{{ item }}" key="{{ lookup('file', 'public-keys/' + item + '.pub') }}"
with_items: list_of_my_users_obtained_earlier
I noticed several syntax issues. Such as using {{ }} inside of other {{ }}, which is not needed or allowed. Missing quotes around your path in the lookup. A missing space between '-' and 'list_of_my_users_obtained_earlier', which is not actually needed, you can just give the list on the same line was with_items. You were using 'item' in one place, but 'item.1' in another.