User `\set` to set set a variable inside psql.  
Read about SQL interpolation in the manual here:

- https://www.postgresql.org/docs/current/app-psql.html#APP-PSQL-INTERPOLATION

Ideally, your script would read:

    SELECT * FROM :"my_schema".:"my_table" LIMIT 1;

With schema and table double-quoted separately, for identifier-interpolation.
Then set schema and table in psql without quotes like:

    my_db=>\set my_schema core
    my_db=>\set my_table product

Just the bare, case-sensitive names.
Finally, execute:

    my_db=>\i my-script.sql

This way, the script is safe against SQL injection.