There does not seem to be much available on the operations around the lake in English, but a very thorough study is available online in Russian - The Great River War, 1918-1920 by Alexander B. Shirokorad (2006) at http://lib.ec/b/385785/read, and most of what follows comes from there. With luck someone might translate the whole book properly in the future.
The Allied base was at the town of Medveja Gora, at the head of the lake. The Allies didn't need an airfield, all they needed was open water - seaplanes had been shipped up to Murmansk on the carriers HMS Nairana and HMS Pegasus and it was easy to fly then down to Onega, which as the 2nd largest lake in Europe made a perfect operating area. Facilities were built rapidly and the first planes arrived on June 6th 1919.
A Fairey IIIC being unloaded from a seaplane carrier
Two types of seaplane were used. The first to arrive were Fairey IIICs. These were 2 seaters, with a pilot and observer, and could carry up to 500lbs of bombs under their wings. In many cases this meant 2x 230lb bombs, specially developed by the British for destroying fortifications, bridges or railways. They also had a forward facing Vickers machine gun, and a Lewis gun on a flexible mount for the observer.
The Short type 184 was another 2 seater, which as well as carrying 520 lb of bombs could also carry a torpedo. What it didn't have was a forward facing machine gun, just a Lewis gun for the observer.
Apparently the Bolsheviks also had an airforce, at least in Febuary 1919 they are listed as having two Grigorovich M9 seaplanes at their lakeside base at Petrozavodsk. In many ways the M9 was equivalent to the British machines, it could even be modified to carry a 37mm cannon, but they seem to have made no impression in the campaign. Perhaps because the armament wasn't available, the itinerary lists only 5 and 10 lb bombs. In any event the Allies had complete air superiority.
A Grigorovich M9
On June 7th, the day after they arrived, two Fairey IIIcs on patrol found and attacked three gunboats, no.s 4, 6 and 8 with 112 and 20 lb bombs. The gunboats, actually converted paddle steamers, were each armed with two machine guns but failed to hit their attackers. On the other hand, although four bombs were dropped on both gunboats 4 and 8, none hit.
On June 9th the Bolsheviks reported bombing raids over a period of 4 hours on gunboats 2 and 7, with apparently 21 bombs dropped on number 2, and 7 on number 7. The British view of this can be seen from the commendation of Observer Officer Frederick Eades, published in the London Gazette.
"On June 9, 1919, whilst serving with the " Syren " Force in North Russia, this officer was on night patrol (observer) in heavy rain, investigating the enemy's position off Schunga, on Lake Onega. Their position was attacked from a height of 300 ft. by machine gun and bombs, the latter being thrown from the observer's cockpit. Having expended all ammunition, the machine returned to base, and after refilling, again attacked and drove one gunboat ashore. A second gunboat was later seen to be in tow, presumably damaged in the attack."
The seaplane base at Medveja Gora
On June 11th Eades was again in cation, flying as observer to Flying Officer Issac. Three Fairey IIIcs made an attack on a gunboat patrol, but although they approached to within 50m only minor casualties were caused.
It was becoming clear to the Bolsheviks that their anti-aircraft fire was pretty useless, even when supplemented by rifle fire. To justify this to their superiors the crews started to report "armoured" bombers, even though the British had nothing of the kind!
Bridges and trains also made tempting targets, though very difficult ones, and to increase accuracy the pilots started to come in low, at about 30ft. A bombing run on the railway on June 11th blew a 6 meter crater in the track using a 230 lb bomb. On June 29th a patrol found came across a train and attacked it directly, coming in at 70ft, and zigzagging to avoid anti-aircraft fire. They dropped a 230 lb bomb directly in front of the train, which ripped the engine in half.
A Short 184 seaplane
Attacks on gunboats were common but generally unsuccessful, although on July 1st the tender "Auguste Blanqui" was hit by a bomb which blew up the bridge, though it still managed to limp back to Petrozavodsk. One innovation was the use of shrapnel bombs, which didn't need a direct hit to work. On June 29th the captain of gunboat 2 was severely wounded by a bomb exploding over 10m away.
By July 10th the Bolsheviks were forced to abandon lake patrols, and the British started to attack Petrozavodsk directly. On August 5th a flight of 4 seaplanes bombed the Bolshevik base there, dropping 20 bombs around the base and the docks. There were further raids on August 7th and then on the 18th, when a 230lb bomb made a crater over 4m deep.
In September the British started to prepare for withdrawal, transferring some of their aircraft to White Russian forces and training up pilots. The last patrol was on September 29th, by Captain Park and a Russian observer, Lt. Motorin.
In total British seaplanes on Onega flew a total of 616 hours and dropped 1,014 bombs as well as 25,000 propaganda leaflets, and shot 47,500 rounds of ammunition. Reconnaissance flights photographed and mapped over 250 square miles.