Adding A Dampening Effect On Camera Following Player?

Currently, I am running a script attached to my main camera that follows my player as he moves along the x axis, but ignores the y axis by staying static.

How could I go about modifying this to incorporate a dampening effect to create asmoother camera transition (very jerky at the moment).

using UnityEngine;
using System.Collections;

public class CameraMove : MonoBehaviour {

	public Transform player;

	void Update () 
	{
		transform.position = new Vector3(player.position.x + 4, 0, -10);
	}
}

Thanks for any help.

Take a look at Vector3.MoveTowards() or Vector3.Lerp(). Lots of posts on UA on both. An example with your script:

using UnityEngine;
using System.Collections;

public class Bug25a : MonoBehaviour {
	
	public Transform player;
	public float speed = 1.5f;

	private Vector3 dest;

	void Start() {
		dest = new Vector3(player.position.x + 4, 0, -10);
	}
	
	void Update () 
	{
		dest = new Vector3(player.position.x + 4, 0, -10);
		transform.position = Vector3.Lerp (transform.position, dest, speed * Time.deltaTime);
	}
}