how great the impact
Well I think there would be very little short term impact in terms of Ottoman offense. The Ottomans really can't advance through entrenched Russian mountain corps, even with a strength 10 better HQ like Sanders. This is true even if the Russians elect to send minimal forces to the Caucasus. The road network is poor especially for the Ottomans who are farther away from rails and have fewer roads to work with.
In terms of Ottoman defense, it would help once it gets to full strength, but the Ottomans would have to prioritize repairing that HQ, very likely starting at strength = 1 due to attack from covid (er I mean typhus). This would take a while, and I'm not sure low-supply and poorly led Ottoman troops would be strong enough to resist mountain corps attacks in the first few turns (would have to test this). I'm so used to them getting better (barely adequate) supply and leadership from the Sanders HQ.
Honestly not sure why to start Ottoman HQ at low strength. Even a full strength Ottoman HQ doesn't make an advance towards Kars possible. The important thing it does is allow you to entrench some extra mountain hexes for leverage (you can get 5+ supply on a couple of of those hexes without roads and thus put up a reasonable entrenched defense). Without decent supply, the Ottomans can't leverage these mountain hexes for defense and the Russian advance towards Erzurum wins up being even easier. I'm not 100% sure a strength 2 HQ would be an acceptable replacement for Sanders and it might be so weak in the first few turns that I would wind up sending Sanders anyhow.
In the long run, Enver would get sacked (historical as he never again commanded an army after his disaster attempting to retake Kars) but probably replaced by the best general (Ataturk) which would be ahistorical. (Or the Ottomans could put Ataturk on top of Sanders or Demjal, depending on which front is "hot"). Pretty much all the manual HQ leader changes in this game are ahistorical so that's a separate issue. Long run the Turks would have an additional HQ (again very expensive for them if you start it at strength = 2).
So what does an extra Ottoman HQ entail? Well, if Gallipoli becomes a legitimate threat, then Sanders may very well stay put in Thrace. OR a daring Ottoman player may elect to rail Sanders down to Egypt and send Demjal Pasha on his long walk towards Baghdad/Basra. In my experience in MP, I started marching Demjal HQ immediately as soon as he spawned upon mobiliztion. (This was in a game where I conquered Serbia VERY quickly so the Bulgarian army was available for defense of the Sinai). Demjal arrived in the Basra area in time to meet a British army which also has a HQ which recently arrived. I believe the British were getting frisky and were beginning to counterattack my low readiness low morale units. The Ottoman HQ boosted the Turk trenches and stalemate around Basra began.