Swipe and hold character movement not working

It only works when you keep swiping because you only update the player's position during the TouchPhase.Moved event. If you want to update the player's position even when the user's finger hasn't been moving, you need a default case:

void Update () {

    foreach (Touch FingerTouch in Input.touches)
    {
        if(FingerTouch.phase == TouchPhase.Began)
        {
            FingerInitialPosition=FingerTouch.position.x;
        }
        else if(FingerTouch.phase == TouchPhase.Moved)
        {
            FingerMovedPosition=FingerTouch.position.x;
            if(FingerMovedPosition>FingerInitialPosition)
            {
                charcter.transform.Translate(Vector2.right * speed * Time.deltaTime);
            }
            else if(FingerMovedPosition<FingerInitialPosition)
            {
                charcter.transform.Translate(-Vector2.right * speed * Time.deltaTime);
            }
        }
        else if(FingerTouch.phase == TouchPhase.Ended)
        {
            FingerInitialPosition=0f;
            FingerMovedPosition=0f;

        }
        else //NEW PART!//
        {
            FingerMovedPosition=FingerTouch.position.x;
            if(FingerMovedPosition>FingerInitialPosition)
            {
                charcter.transform.Translate(Vector2.right * speed * Time.deltaTime);
            }
            else if(FingerMovedPosition<FingerInitialPosition)
            {
                charcter.transform.Translate(-Vector2.right * speed * Time.deltaTime);

            }
        }
    }
}

In addition to the new statement, the others are all else if as well.