You wouldn't expect to run a marathon after one day of training, and you shouldn't expect a supplement to change your physiology overnight
On a GLP-1 medication, that assumption can work against you
Glucagon receptor activation increases metabolic rate by stimulating thermogenesis in brown adipose tissue and enhancing mitochondrial activity in hepatocytes and skeletal muscle
M., Bachmann, P., Safi, T., Pfeiffer, N., Gericke, A
For L-carnitine specifically, most healthcare professionals would not routinely recommend supplementation for people taking Mounjaro unless there is a documented deficiency or a specific clinical indication