Im not 100% sure but I don't think that the 8 bit pngs in PS works very well with opacity, and therefore if your sprites in 24 bit depth have smooth edges, they will render a white border around the image where the opacity occurs.
To solve it you have some options:
1. Make all sprites with hard edges, mean you paint them with pixel setting in PS. This is the time consuming way, as you will have to repaint all the sprites.
2. You can use this website to convert 24 bit to 8 bit pngs and will preserve the opacity as well. You can only do it with one image at the time so if you have a lot it will ofc take some time as well. To save the 8bit image you just right click the image it self and choose "save image as.."