Error after second read from CTE or TEMPORARY TABLE in Postgres

I am trying to write a query where I select records by some condition, update one column of those records and then return them, all in one transaction.

I have tried two options. First with CTE:

BEGIN;
WITH unwatched_rows AS (
        SELECT * from my_table WHERE is_readed = false FOR UPDATE
)
UPDATE my_table SET is_readed = true WHERE id IN (SELECT id FROM unwatched_rows);
SELECT * FROM unwatched_rows;
COMMIT;

In this case I get an error relation "unwatched_rows" does not exist. How I learned this is because i can't use CTE more than once. And as I understand it, CTE is not suitable for my situation

Second option with CREATE TEMPORARY TABLE:

BEGIN;
CREATE TEMPORARY TABLE unwatched_rows AS (
        SELECT * from my_table WHERE is_readed = false FOR UPDATE
);
UPDATE my_table SET is_readed = true WHERE id IN (SELECT id FROM unwatched_rows);
SELECT * FROM unwatched_rows;
COMMIT;

I am running this query through DBeaver. For first time it works fine, but when I try to run this query again I get an error relation "unwatched_rows" already exists. I don't understand why this is happening. Shouldn't the temporary table be dropped right after the commit?
What am I doing wrong and how to solve it?

If you want the temporary table to be removed right after the commit, You will have to specify ON COMMIT DROP when create it :

BEGIN;
CREATE TEMPORARY TABLE unwatched_rows ON COMMIT DROP AS (
        SELECT * from my_table WHERE is_readed = false FOR UPDATE
);
UPDATE my_table SET is_readed = true WHERE id IN (SELECT id FROM unwatched_rows);
SELECT * FROM unwatched_rows;
COMMIT;

An other solution is to drop your temporary table before created it again :

BEGIN;
DROP TABLE IF EXISTS unwatched_rows;
CREATE TEMPORARY TABLE unwatched_rows AS (
        SELECT * from my_table WHERE is_readed = false FOR UPDATE
);
UPDATE my_table SET is_readed = true WHERE id IN (SELECT id FROM unwatched_rows);
SELECT * FROM unwatched_rows;
COMMIT;

You can certainly use a CTE (with ...) once. However you used it incorrectly. A CTE is essentially a view which exists only for the duration of the query where it is created. In query the semi-colon (;) after the update terminated the statement and thus the CTE itself goes out-of-scope. So for theselect statement it no longer exists. Postgres allows DML in a CTE so to correct your statement create a second CTE that does the update. So:

with unwatched_rows as (
        select id 
          from my_table 
         where is_readed = false for update
     )
   , upt as (
       update my_table 
          set is_readed = true 
        where id in (select id from unwatched_rows)
     ) 
select * 
  from unwatched_rows;

As far as wrapping into a function this is actually quite simple. It does not require a temp table not, for that matter, a CTE. It reduces to a single SQL statement:

create or replace function unwatched_rows()
  returns setof my_table.id%type 
 language sql 
as $$ 
    update my_table 
       set is_readed = true
     where not is_readed    -- same as: is_readed = false
    returning id;
$$;

See here for demo illustrating both.