There are no reports from the 13th cavalry division, luckily however all is described quite well in reports of the 15th cavalry division. Further I translated most important parts of that report:
At 01:00 an order was received to be ready to attack Kuschevsky at sunrise in cooperation with 2 regiments of 13th cavalry division and a separate tank brigade (also called Maikop tank brigade sometimes) and 1 regiment of 12th cavalry division.
The forces arrived at around 06:00 02.08.1942. Before that, at 04:00 a company of enemy SMG-men (infantry) took Veseliy village.
In Kuschevsky there was an enemy 4th Mountain infantry division, reinforced with cavalry, artillery and tanks. (...)
The plan was as follows:
13th Cav. together with TBr (Tank Brigade) had to move to the left flank of the 25th cav. regiment (...) with tanks in front. (...)Infantry, mortars, artillery and tanks should provide as much fire support as they can to suppress the enemy, while the cavalry with tanks moves to the village Vesely. Simultaneously, the 15th cavalry division should begin their attack on Kushevskaya on foot.
From 06:00 till 10:00 13th Cav. was performing recon missions, to find targets for the artillery to suppress.
At 10:45 the attack began.
Tanks, supporting 13th Cav. were moving too slowly and were stopped, so 13th Cav. was forced to move on its own. 15th Cav. was hammered by a bomber.
25th cav. Regiment failed to install their artillery quickly enough to support the 13th cav. and fully suppress the enemy. Nevertheless, they did reach their objective and forced the enemy to retreat and take Vesely.
They lost 260 men during this assault.
15th cav. Took part of Kuschevskaya and the 25th cav regiment entered the city and continued the fight near the station. 42d cav regiment while approaching from the southeast engaged in a melee combat.
All these actions forced the enemy to retreat.
15th cav. lost 200 men.
Further assault was impossible because of poor actions of TBr. and lack of reserve (13th cav was the reserve, which was already in use). Because of impossibility to exploit their success and inability to quickly prepare defensive positions, they decided to retreat back to their previous positions.
Summing up the 13th cavalry division, with tanks in front and covered by fire from artillery of the 25th cavalry regiment performed a quick flanking manuver, unmounted and took the village Veseliy. At the same time the 15th cavalry division stormed Kuschevskaya on foot from the front. As a result they took Kuschevskaya but shortly after retreated, because did not have a proper force to defend the village. Both divisions in total lost 460 men.
So horses were used here to do a quick flanking maneuver to put the defenders of Kuschevskaya into danger of being surrounded inside the city, thus forcing them to retreat. Fighting was done on foot, as it was prescribed in the Red Army manuals. No one sane would charge onto a tank. So they would use horses solely for transport and unmount as soon as they engage serious force. Such small details however are not described in the report.
Note that tanks are in front and cavalry is behind. Tanks are faster than infantry and usually there is a problem, when tanks go too fast and arrive at location without infantry cover, where they are immobilized and destroyed by enemy anti tank teams. Cavalry in this case acts as a replacement to APCs (which the Red Army lacked) for the infantry to keep up with faster tanks. Charging tanks were never valid tactics at that time.
Original: report of the 15th Cavalry division (keeps opening on the wrong page for some reason, but it is there).
17th corps report.
Scheme of actions of 15th Cavalry division on:
01.08.1941, 02.08.1942, 03.08.1942
Note2: 4th corps was disbanded in 07.07.1940 so it did not take part in the Great Patriotic war.
P.S.: The "Unwomanly Face of War" is not a history book, but a fiction book.