Normally when the ESC is first turned on, it expects to see the throttle at one position, then as you move it down to full idle, it "learns" the throttle range and arms the ESC.

If the ESC is not able to read the throttle range, moving the stick can result in a limited low range which in turn results in what you are seeing.

You may need to be sure that your TX is set up for maximum throttle "throws" or range too.

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If the ESC's timing is not set up properly for the motor or for the wrong cell count, you will also experience what you are seeing....

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If there is a broken or shorted winding on the motor, you will also get the same result.

You could check each of the three connectors from the motor with a meter to see if this is the case.

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Of course if the battery is not putting out the proper voltage the ESC could be cutting out power to the motor too.

PM John here with all of this.