I did find this detail on a Turkish webb site;
"Early in the morning on February 16, 1916, Russian troops marched into Erzurum. Turkish units had successfully withdrawn and avoided encirclement, however casualties were already high. 327 pieces of artillery were lost to Russians. Support units of the Third Army and around 250 wounded lying at the hospital of Erzurum were taken prisoner. Meanwhile remnants of the X and XI Corps established another defensive line, 8 kilometers east to Erzurum.
Istanbul was still celebrating the victory in Gallipoli. However, news from the eastern front and the loss of Erzurum changed the atmosphere in an instant. Enver Paşa ordered the V Corps (consisting of 10th and 13th Divisions) to be deployed to the Caucasian front. On February 27, he replaced Mahmut Kamil Paşa with Vehip Paşa, then commander of the Second Army and one of the heroes of Gallipoli. Vehip Paşa arrived in Erzincan, the new location of the army headquarters, on March 16. His first task was to bring an order to the Third Army, which at that time had only 25,500 men, 76 machine guns and 86 pieces of artillery battle ready. Erzurum was lost, together with its hospitals and logistics support.
Enver Paşa knew that the Third Army was not capable to defend the frontline on its own. He ordered the Second Army, which was then in Thrace under the command of Ahmet İzzet Paşa, to proceed to Diyarbakır in Southeast Anatolia and support the eastern flank of the front held by the Third Army. The Second Army was planned to reach the strength of four corps and ten divisions by August.
Meanwhile Russians were moving full speed ahead. In March 1916, they landed in Rize, an important port in eastern Black Sea Region, marching to west and occupying another port, Trabzon, on April 16. This was bad news for the Third Army, because now they were cut off from reinforcements and supply through Black Sea. In addition to those difficulties, the arrival of the Second Army was delayed, because there were limited means for railroad transportation and it was the units deployed to Mesopotamia who were given priority in transportation.
Facing so many difficulties, Vehip Paşa decided to divide the front in three operational zones: (1) Southeastern zone: North of Diyarbakır, to be held by Mustafa Kemal’s XVI Corps; (2) Central zone: Commanded by X Corps’ Yusuf Ziya Paşa, supported by IX and XI Corps as well as the 2nd Cavalry Division; (3) Northern zone: Black Sea coast, to be held by Fevzi Paşa’s V Corps.
Russian forces entering Trabzon
In late June, Fevzi Paşa’s V Corps attacked into the Eastern Black Sea Mountains (Pontic Alps) in order to recapture the port of Trabzon, which was now used by the Russian for seaborne reinforcements. Although minor successes were registered, the main objective was now achieved due to lack of adequate forces. By June 28, the Turks were ten kilometers to the sea, however they had to stop there due to a strong Russian defense.
Meanwhile the Russians were preparing for a counteroffensive aiming to relieve the pressure on Trabzon and threaten the Turkish city of Sivas. They launched their attack, which came to be known as the Çoruh Campaign, on July 2; and as they were on the outskirts of Bayburt, they engaged the Turkish X Corps. The Turks fought bravely, but they could not hold the ground. Bayburt fell on July 17.
Vehip Paşa and the staff of the Third Army
Source: Atlas Tarih 30/2014 (above), Tunca Örses collection (below)
The Russians did not stop at Bayburt. Using the town as a bridgehead they renewed their attacks, crossed the Karasu River, pushing back the Turkish IX and X Corps. On July 25, Russian advance forces entered the city of Erzincan. Vehip Paşa had no option but to leave the town to the Russians and retreat to west in order to prevent further Russian penetration into Anatolia. Çoruh Campaign had gone on for 12 days during which the Turks have not only lost important towns, but also 17,000 men killed and around the same number taken prisoner.
Although the Turks had received a remarkable blow, Ahmet İzzet Paşa decided to attack one week after the conclusion of the Russian offensive. It was now the Second Army’s task to save the Third Army from disaster and recapture the town of Erzincan. The Turkish offensive commenced on August 2, 1916, in three corps-sized groups, III, IV and XVI Corps. In the earlier periods of the campaign, Mustafa Kemal’s XVI Corps managed to take Bitlis and Muş, however this initial success did not bring victory. Russians were strengthening their lines and two weeks after the launch of the Turkish offensive they were strong enough to respond with counteroffensives. At the same times, the Turks were suffering from severe supply and logistics problems. By late September, the Turkish attack was finished. During these two months, the Turks had gained some ground, at the cost of around 30,000 killed and wounded.
The rest of the year 1916 was spent by the Turks with organizational and operational changes in the Caucasian front. Fortunately for the Turkish commanders, the Russians were quiet during this period. The winter of 1916-1917 was extremely harsh, which made fighting nearly impossible. This situation did not change during the spring. Meanwhile Russia was in political and social turmoil, which was also influencing the army ranks. The chaos caused by the October Revolution put a stop to all Russian military operations and the Russian forces began to conduct withdrawals. Neither the Russian soldiers nor the Russian people wanted to go on with the war anymore. The Turks on the other hand, could not take advantage of this situation; since their units were not in good shape either. They were under great pressure from the British in Palestine and Mesopotamia, therefore withdrawing the majority of their forces (five divisions) and sending them south. The year 1917 passed by without fighting in the Caucasian front. The Russian army slowly disintegrated until there was no effective military force. The Armistice of Erzincan, signed on December 16, 1917, officially brought an end to the hostilities.
This covers that fighting the best I can find.
The 2nd Corps is not mentioned, so was possibly to beef up units of the 3rd Army which suffered heavy losses over the last months around Erzurum (all three Corps 5th, 10th and 11th were disbanded soon after) or was in 2nd Army reserve?
Considering the length of the front at that time, these Corps (3rd and 4th) must have supported Kemals (16th) Corps in its attack around Bitlis and Mus?
But I can find no other details on this fighting or the losses they had?