While Home Assistant can both send and receive MQTT payloads, MQTT is not the normal method for communicating internally. If you want, you can set up an automation with an MQTT trigger and the desired climate service call(s) using whatever topic and payload you want.
How about publishing via mqtt the current status of the heat mode?
the reason I want to do it via mqtt is cause I have a small LCD on the wall which I want to give control of the thermostat by pressing a button in the lcd. this lcd is a esp32, so I need mqtt
But to control something in HA from an external device using MQTT payloads you need the opposite, you need an automation that triggers on receipt of an MQTT message and calls the appropriate climate service.