Why does electronegativity generally increase across a period?
Electronegativity is the tendency of atom to attract the electron cloud towards itself in a covalent bond. Across the period, since the electrons are filled in the same shell(which means it should be of the same size without any electron) and the amount of positive charge increases and as a consequence the size of the atom also decreases(Each electron is pulled more strongly by a nucleus with higher positive charge). The electrons are hence more tightly held as you go across the period.


  I thought though it was because as you go across a period, an atom
  will slowly fill up its outer shell, and hence reduce its need to gain
  or hog electrons?? Or is this incorrect?


A need to gain or lose electron is dictated by the nearest configuration of greatest stability(Usually Octet). And this is not closely related to Electronegativity either. Electronegativity only has to do with how strongly can an atom pull the electron cloud towards itself. Actually, as you go across the period, the tendency to gain electron increases(Indicated by increase in Electron Affinity) and not decrease as you thought.