I don't think a bad battery will cause these codes, I would have the battery load tested at the auto parts store and then replace it.
P0036 - HO2S Heater Control Circuit (Bank 1, Sensor 2)
P0135 - HO2S Heater Performance (Bank 1, Sensor 1)
P0304 - Cylinder 4 Misfire Detected
I would replace both oxygen sensors because the heater is no good on both of them. Then you have a simple misfire problem with Cylinder 4, so remove the spark plug to see if the cylinder has compressor and then replace the plug and the ignition coil. This guide can help walk you through how to fix an engine misfire:
https://www.2carpros.com/articles/engine-misfires-or-runs-rough
Here are the locations of the oxygen sensors. Check out the images (below). Let us know what you find.
Images (Click to make bigger)
Saturday, January 11th, 2025 AT 9:28 AM