You are right in that adding acid increases the concentration of dissociated ions in water, and that the hydrogen goes to the cathode but you have assumed too much with the oxygen part.
In electrolysis/electroplating, the positive ions move towards the negative cathode and are reduced, and vice-versa, the negative ions move towards the positive anode and are oxidized. For a pure acid solution (no salts), hydrogen is the only postivie ion so it migrates towards the cathode, however not only do the hydroxide ions move towards the anode, but so do the conjugate bases of the acid. These conjugated bases may or may not be oxidized before the hydroxide depending on potentials and available mechanisms for reaction to occur.
For example chlorine can oxidize before oxygen and electrolysis with chlorides forms aqueous chlorine as shown $\ce{2Cl-_{(aq)} -> 2e- + Cl2_{(aq)}}$ and sulfate can form persulfates at the anode: $\ce{2SO4^2- -> (SO3OOSO3)^2- + 2e- }$. These bases compete with the $\ce{OH-}$ in solution to be oxidized. These species also tend to be very reactive and will often oxidize the anode unless an inert material such as graphite or platinum is used (sometimes nickel will work).
In electroplating, the metal ions compete with the hydrogen ions to be reduced and indeed hydrogen gas is formed at the same time that electroplating occurs. for ever $\ce{H+}$ reduced there is a $\ce{OH-}$ left in solution. in electrolysis the electron to reduce the $\ce{H+}$ would come from the $\ce{OH-}$, but in electroplating it comes from the metal (i.e. $\ce{Cu -> Cu2+ + 2e-}$). NOw the elctroplating process has created $\ce{Cu2+}$ ions and left $\ce{OH-}$ ions in excess which creates $\ce{Cu(OH)2}$ which precipitates creating anode mud.
This is where the acid comes in to keep the bath clean, I'll use sulfuric acid as an example. The sulfuric acid prevents the formation of $\ce{Cu(OH)2}$ by providing $\ce{SO4^2-}$ and $\ce{H+}$ ions to react with the $\ce{Cu(OH)2}$ and keep the copper in solution ($\ce{Cu(OH)2 + H2SO4 -> CuSO4 + 2H2O}$).