How to move the camera closer to the player while there is an object between them?

Here is a bit of addition to your code. Not sure this will be enough when you get among buildings. You may have to raycast multiple times to prevent clipping on some building, but anyway this gives you a start.

 using UnityEngine;
 using System.Collections;
 [AddComponentMenu("Controls/CameraControls")]
 public class CameraControls : MonoBehaviour {
 
 public Transform player;
 public float unobstructedDist = 8.0f;
 public float distSpeed = 1.25f;
 
 public float xSpeed = 90;
 public float ySpeed = 90;
 
 public float yMin = -20;
 public float yMax = 80;
 
 private float x = 0.0f;
 private float y = 0.0f;

 private float distance;
 private float desiredDist;
 
 
 void Start () {
     distance = unobstructedDist;
     desiredDist = unobstructedDist;
     Vector3 angles = transform.eulerAngles;
     x = angles.x;
     y = angles.y;
 }
 
 void LateUpdate () {
     DistanceAdjustment();
     distance = Mathf.Lerp (distance, desiredDist, Time.deltaTime * distSpeed);
     if(player) {
         x += Input.GetAxis("Mouse X") * xSpeed * 0.02f;
         y -= Input.GetAxis("Mouse Y") * ySpeed * 0.02f;
         
         y = ClampAngle(y, yMin, yMax);
         
         Quaternion rotation = Quaternion.Euler(y,x,0);
         Vector3 position = rotation * (new Vector3(0.0f, 0.0f, -distance)) + player.position;

         transform.rotation = rotation;
         transform.position = position;
     }
 }
 
 void DistanceAdjustment () {
     RaycastHit rayHit;
     if(Physics.Raycast(transform.position, transform.forward, out rayHit, 15)) {
         if(rayHit.collider.tag == "Buildings") {
             desiredDist = unobstructedDist - rayHit.distance;
         }
         else {
             desiredDist = unobstructedDist;
         }
     }
 }
 
 static float ClampAngle (float angle,float min, float max) {
     if(angle < -360) {
         angle += 360;
     }
     if(angle > 360) {
         angle -= 360;
     }
     return Mathf.Clamp (angle,min,max);
 }
 }

I've added 'unobstructedDist' and 'desiredDist' and a bit of code that Lerps between current distance (distance) and desiredDist. Note your Raycast was using Vector3.forward and needed to use transform.forward.