Step 1: Identify the article.
The question is about Article $39A$ of the Indian Constitution and what it deals with.
Step 2: Where this article sits.
Article $39A$ is a Directive Principle of State Policy. These principles guide the State to make laws for the welfare of the people.
Step 3: What Article $39A$ says.
It directs the State to give equal justice to all and to provide free legal aid, so that no person is denied justice just because they are poor.
Step 4: Link to legal aid bodies.
To carry out this idea, India set up Lok Adalats. These are people's courts that settle disputes quickly, cheaply, and through friendly settlement.
Step 5: Test the other options.
It is not a Fundamental Right, it is not about local self government, and it is not about the powers of the Supreme Court. So those choices do not fit.
Step 6: Pick the best match.
Since Article $39A$ is the base for free legal aid and speedy justice, it is most directly connected with Lok Adalat.
\[ \boxed{\text{Lok Adalat}} \]