Can being sick with a cough cause a lump in throat?

Yes: A cough is likely a part of an upper respiratory tract infection, probably viral in nature. If the lump is inside your throat this could also be a strep tonsil infection. I would first recommend rest, plenty of fluids and you can also try gargling with salt water. If you get worse or not much better in a few days, I would seek medical advice. All the best.