MySql - Read only on slave
Here are the grants you just mentioned

GRANT RELOAD, PROCESS ON *.* TO 'my_app'@'%' IDENTIFIED BY PASSWORD '*'
GRANT ALL PRIVILEGES ON `my_app`.* TO 'my_app'@'%' 


Based on this, you should not be able to write to the Slave because SUPER is a system level privilege (SUPER only appears in mysql.user as Super_priv) and not a DB level privilege (SUPER does not appear in mysql.db). The first GRANT lacks SUPER privilege. The second GRANT does not have a SUPER privilege context at all.

Having SUPER privilege has a lot of firepower because you can run the following


CHANGE MASTER TO
KILL 
SET GLOBAL
PURGE BINARY LOGS


Not every user needs this. Giving the SUPER privilege to just anyone can actually hamper a DBA from logging into mysql if max_connections is reached.