Pass object name as variable to a script

I am sorry to have to ask this question but I do not understand why this isn’t working!
I have declared the following in my script:

          public string Markername;

and it appears in the editor inspector
When I put the name of the object in this field and then run the game, the string isn’t passed to my script. It remains blank:

         Debug.Log( "object name" + Markername);

please tell me what have i done wrong…

Full script:

using System.Collections.Generic;
using UnityEngine;
using UnityEngine.SceneManagement;
using System.Collections;

public class Gazeandwalk : MonoBehaviour
{
    public bool mousehover = false;
    public float Counter = 0; 
    public string Markername;

     void Start()
     {
         Debug.Log("starting walking script");
         Debug.Log("object name in startup: " + Markername);
         
     }
    
     // Update is called once per frame
     //void Update(string Markername, float Counter)
     void Update()
     {
         
      
         if (mousehover)
         {
             var pos = GameObject.Find("Sphere").transform.position;
             Counter += Time.deltaTime;
             if (Counter >= 3)
             {
                 mousehover = false;
                 Counter = 0;
               //move player
                    Debug.Log("should be walking to position " + pos + "object name in update: " + Markername + " Counter " + Counter);
                         
               transform.position = pos;
                 
             }
         }
     }
    
     public void MouseHoverChangeScene()
     {
         Debug.Log("setting walk");
         mousehover = true;
     }
    
     // mouse Exit Event
     public void MouseExit()
     {
         Debug.Log("cancelling walk");
         mousehover = false;
         Counter = 0;
     }

 
}

thanks for helping me with this - simple but stuck!

Call it in the Update and see what happens.
And also, avoid starting variable names with cap letter. Use camel case instead, like markerName. Start with cap letter when creating classes and functions

thanks @LeonmFF
that is the problem - when I reference in update() in debug.log nothing appears
as for variable names - i read that if i want to expose the variable in the Inspector i need to start with a capital?