13s gives you your voltage and you match this to your controller and motor. 13 x 3.6v/cell = 46.8v or nominally 48v
10p or what ever *p gives you your amp/hour depending on the cells used. In a way think of this as your gas tank. It doesn't matter to a motor if you have a 10 gallon or 15 gallon tank. The bigger tank just gives you longer drive time.
The controller amp is what governs the maximum amount of power delivered to the motor. It should be matched to the motors capabilities.
That said... Larger motors coupled with the higher current controllers should be matched with greater output (more cells in parallel) batteries as they will draw more more power. 250w vs 500w
Harry's answer above is correct about your problem... Poor quality cells and/or not actually 6p.
My current battery as an example.
Battery Cell: Samsung 35ET 3500mah.
Array mode/Capacity : 13s5p: 48v / 17.5AH
I have no problem using this battery with a 750w/25a - motor/controller with minimal sag
Numbers are usually rounded up by battery builders so keep this in mind.
EDIT: One factor often overlooked is the battery Discharge Current. This is the batteries ability to deliver power continuously.
The Samsung cell is capable of delivering 8a continuously... So in a 5p battery this results in 40a continuously, hence the minimal sag. This is where many of the lower quality cells fail and why they should be avoided. You can think of this as the ability to delivery gas to an engine. Dating my self here but... a 2 barrel vs 4 barrel carburetor on 1960's cars.
When building a battery my opinion is to over build as far as capacity and current as this will keep the cells from stressing (heating up, a cells #1 enemy) and be more viable as it ages