2002 mitshubishi montero sport 3.0L

My 2002 montero sport 3.0L gets a p0403 err and idles rough in park and even rough in drive at at stop. I notice my egr valve would move a little and stay in that position at idle and close back after I turn off car. When I remove the vacuum hose from the egr valve the engine would idle better. I found that my egr vaccum solenoid wouldn’t be energize so I disconnected the wire harness to the vacuum solenoid and ran test I found that hooking it straight to the battery it would click. I then check the wire harness and found that when I put my pos lead to one side the connector and touch the battery neg I had 12VDC, then I took the neg lead and tough the harness neg and tough the bat pos and I had 10.5VDC. I then retest the harness at the egr vacuum solenoid and had 10.5VDC. When I plugged it in to the solenoid the voltage went away. I check the ground back to the ECM it check good no break and not shorted. I did observe that the solenoid would click for a split second and then turn go off. What other issues could cause the egr vacuum solenoid not to function? Is there any other components that might cause the ECM to cut output to the vacuum solenoid.