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get the most frequently appearing user_code where the month is specified

Since user_code is the primary key, that question would be nonsense. There can never be more than one. I assume you meant invite_code?

Just add a WHERE clause. And 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;

Month, date, timestamp?

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'

The column could look like this (also addressing your comment):

...
, inserted_at date DEFAULT CURRENT_DATE
...

The default value is entered when the column is omitted in an INSERT statement.

Or, if really only the month is relevant:

... DEFAULT date_trunc('month', now())::date

Or store the complete timestamptz (that's what I would probably do):

, inserted_at timestamptz DEFAULT now()

Read the manual here and here.