Reaction between barium peroxide and hypochlorous acid
You may have confused barium peroxide with barium metal.  The metal can indeed displace hydrogen from hypochlorous acid, not to mention (for a metal as reactive as barium) from the water in which the acid is dissolved; but metal peroxides will produce oxygen and water from the acid.  So your missing product is water, not hydrogen.
Tricky half-reactions
If we want to balance this with half-reactions we need to take some care, because we need to simultaneously cancel protons and electrons with a weak acid like hypochlorous acid.  Given the reduction half-reaction based on chlorine being reduced from the acid:
$\ce{HClO + H^+ + 2 e^- -&gt; Cl^- + H2O}$  Eq. 1
We must set up the peroxide oxidation to oxygen so that the electron/hydrogen ion ratio is also $2:1$.  Thus a proposed reaction
$\ce{BaO2 -&gt; Ba^{2+} + O2 + 2 e ^-}$  Eq. 2
would be balanced but you don't have the proper ratio of electrons to protons matching the weak hypochlorous acid reduction.  Add water to get the necessary hydrogen ions, and note that this means you change the oxygen production:
$\ce{BaO2 + H2O -&gt; Ba^{2+} + 2 H^+ + (3/2) O2 + 4 e ^-}$  Eq.3
In other words, water is also oxidized to generate the hydrogen ions necessary to effect the reduction of the weak hypochlorous acid, thus producing that extra half-mole of oxygen per mole of barium peroxide.
Four electrons to two protons in Eq. 3 matches the ratio of two electrons to one proton in Eq. 1.  So now we can double Eq. 1 and add to Eq. 3 getting the reaction that avoids both unbalanced protons and unbalanced electrons.  As we saw above, one mole of barium peroxide gives 1.5 moles of oxygen in the overall reaction:
$\ce{BaO2 + 2 HClO -&gt; Ba^{2+} + 2Cl^- + (3/2) O2 + H2O}$.