See the Wikipedia article for more information.
Here is a brief summary:
The CrimeanWar (1853-56) was fought between Russia and an alliance Britain, France, the Ottoman Empire (Turkey) and Sardinia. General causes were conflict between Britain and Russia over control of Dardanelles. Pretext was Russian-French quarrel over guardianship of Palestinian holy places. Turkey's rejection of Russian demands prompted Russia's occupation of Moldavia and Wallachia. Turkey declared war (1853).
France and England joined (1854). Main campaign centering on Siege of Sevastopol in Crimea, was marked by futile gallantry (e.g. battle of Balaklava) and heavy casualties. The Congress of Paris led to the Treaty of Paris on 30th March 1856. This checked Russion influence in SE Europe.
John Capper’s service in Turkey from 4th April to 16th June 1856 followed immediately on the Treaty of Paris. It was presumably part of peace-keeping activites by Britain in support of the Ottoman Empire after the war. It does not seem likely that John was involved in any fighting here. (Detailed research of the activities of the Royal Artillery in Turkey in 1856 might unearth more details.)