Get a "Yellow Jacket" brand cord. I have four of them, each 100' long. They are construction grade, are 12 awg and are rated at 15 amps. I have a feeling the instruction manual was probably written in the same language as the country the mower was built in, and therefore a translation error is causing the manual to state a maximum of 10 amp rating for the cord.
Think of a cord as a water hose, and think of the electric mower as a spray nozzle. Assume a particular spray nozzle has a 4 gpm flow rating @ 50psi. Please don't check my math, I'm just pulling numbers out of the air for discussion purposes. Let's assume a 1/2" diameter, 50' long hose will flow precisely 4 gpm @ 50 psi. With that hose, the nozzle will produce 4 gpm. Put a 5/8" diameter hose on that nozzle that will flow 7 gpm. Guess what, that nozzle will still only produce 4 gpm. Using too large of a cord with your mower is just like using a larger diameter water hose. However, switch that hose to a 3/8" diameter one that only flows 2.5 gpm, and you'll only get 2.5 gpm from the nozzle. Doing this to an electric motor will burn it up, overheat the cord, start a fire, or possible all three.
Think of electric flow in terms of water flow. The wire gauge is the hose diameter. Amperage is the same as water flow in gallons per minute. Voltage is similar to pressure in the sense that you have to increase it as the wire gets longer to account for voltage drop.
Here is a simple calculator I found on the net.
Voltage Drop Calculator JavaScript