I'm running a code in which the IN Pin of a single channel relay is connected to D6 of the Arduino Uno board, and I'm sending High and Low voltage to D6 in between a delay of 1 sec.
The system is working fine when I connect the COM port of the relay to 5V and NC port to an LED, but if I replace the LED with a 5V DC pump, the system doesn't work as per the code...
This is not very surprising. What is the motor specification?
Have you taken any measures to suppress the interference a motor will generate? Motors generate a lot of interference and will often produce enough to reset an Arduino.
Please supply a schematic of your system and a photograph of your setup.
Posting an annotated schematic as requested will save a lot of guessing and questions as well as the time of many of the volunteers trying to help. It appears as a hardware problem so include links to technical information on each of the hardware devices.