Be wary of the power output claims, especially from Chinese sources. Almost universally, for the higher output units, the Chinese now quote power as Input power, so a module quoted as 15Watts is actually around 5 Watts output power. AFAIK the most powerful single laser diode is 7.5 watts output power. If the advert claims more than that they are either pulsing the diode with a current much higher than the maximum current for that diode, which it is widely accepted will shorten its life, or there is more than one diode in the module (like this example, the cost of which is around $250, if you are looking at a substantially lower price then alarm bells should be going off! Many of the advertised as 15 watt modules will have around 3.5 Watts output power and cost around $50.
Looking at the burn marks at the start of your cutting operation, they could be down to Z commands in your gcode causing the XY axis to wait while the un-implimented Z axis to move, or perhaps you are not using GRBL’s dynamic laser power mode .
Which air pump did you go for?