Usually, there's a fuse specifically for heated seats that you can pull to deactivate them temporarily. It's often listed in your vehicle's manual or fuse box diagram. That could be a quick fix until you can get it properly checked out.
Based on what you've described, it could be a fuel delivery problem, maybe related to air getting into the system or a weak fuel pump. The fact that pumping the hand pump helps temporarily suggests it might be related to fuel pressure or air in the lines.