Oh no! The poor thing. This used to happen to me every single time I got an eyebrow wax (Frida Kahlo style natural monobrow here).
I found that using a toner on the area after waxing helped close up the pores a bit and stopped them getting so full of gunk. Drinking lots and lots of water never hurts either.
Whatever you do, make sure whatever you put on his back is as gentle as possible. His poor skin is already irritated as all getout. I’d go for super gentle formulas and stay away from the harsh no-frills soap that boys usually use. (My fella loves his Imperial Leather soap, no matter what I tell him about how harsh it is on my skin…)
Hope it clears up soon!