How is the 10-pin ICSP connected?
The AVR-ISP mkII does not supply power to your circuit. As such, it is incumbent on you to supply power to that pin in-circuit so that the programmer can sense the target voltage. Cutting the connector is an aggressive approach... I would advocate that you just buy a 2x5 male header and solder wires to it.

I have to modify this answer back, I thought it was talking about the Atmel product, not the Olimex clone. From the datasheed:

5.3.1 TARGET jumper
TARGET jumper controls the powering of the target board. If it is in position ON (check the diagram on the back of the plastic cover) it will provide either 3.3V or 5V to the target board (depending on the position of the POWER jumper) The default position is OFF. 

As always read the manual first