I’m struggling getting my score counter to go up only when pressed on the sprite that has collision over it not just anywhere on the screen.
If your sprite has a CollisionObject2D, like Area2D, CharacterBody2D, RigidBody2D or StaticBody2D, you can override its _input_event() method and put your score counter logic there.
Note: _input_event() requires input_pickable to be true and at least one collision_layer bit to be set.