With one equivalent of HBr, R−C≡CH gives a vinyl bromide: Markovnikov puts the Br on the more substituted carbon. With excess HBr the reaction runs again, and here is the part students get wrong: both bromines end up on the same carbon, giving R−CBr₂−CH₃, a geminal dihalide.
Markovnikov twice in a row puts both halogens on the same carbon: geminal, not vicinal.
The reason is the second protonation. It happens in the direction that gives the more stable cation, and the carbon already bearing a bromine can stabilize the positive charge with a bromine lone pair. So the second bromide attacks that same carbon. Geminal is the mechanistic consequence of Markovnikov applying twice, not a separate rule.
Halogenation behaves in a parallel way. One equivalent of Br₂ or Cl₂ gives the dihaloalkene, with anti addition placing the two halogens on opposite sides. Excess halogen adds across the remaining π bond as well, giving the tetrahalide with all four halogens on the two original alkyne carbons.
Oxidative cleavage sits at the far end of this section. O₃ followed by workup, or hot concentrated KMnO₄, snaps the triple bond entirely and gives two carboxylic acids. If the alkyne was terminal, that end carbon has no substituent to keep, so it leaves as CO₂ and you recover only one acid.