> get the most frequently appearing user_code where the month is specified

Just add a `WHERE` clause:

    SELECT user_code, COUNT(invite_code) AS counted
    FROM   invite_table
    WHERE  month = 'May'  -- or whatever is stored in your varchar(3) column
    GROUP  BY user_code
    ORDER  BY counted DESC, user_code  -- to break ties in deterministic fashion
    LIMIT  10;

You would `GROUP  BY user_code` of course, not by `invite_code`.

OTOH, since `user_code` is the PK, the whole question is nonsense. There can never be more than one. Maybe you meant `invite_code` after all? In this case, since the column can be NULL, also consider excluding NULL values:

    SELECT invite_code, COUNT(*) AS counted
    FROM   invite_table
    WHERE  month = 'May'  -- or whatever is stored in your varchar(3) column
    AND    invite_code IS NOT NULL -- exclude NULL
    GROUP  BY invite_code
    ORDER  BY counted DESC, invite_code  -- to break ties in deterministic fashion
    LIMIT  10;

A month column as `varchar(3)` doesn't seem very useful if there can be data for more than a single year. I would use data type `date` for it. You can format that any way you like for presentation. Like:

    SELECT to_char(date '2017-12-01', 'Mon');  -- 'Dec'