Three Laws

The story I was thinking of, "... That Thou Art Mindful of Him" does not involve weakening the laws at all. You might be thinking of "Little Lost Robot" in which a robot with a dangerously incomplete Law One is hiding among otherwise identical robots.

In "That Thou Art Mindful" the robots are given more powerful abilities to make decisions based on the laws in order to be suitable for use on Earth, where they could encounter complicated scenarios with conflicting orders, simultaneous dangers and so on. So rather than being "hard coded" with a description of what makes something human, they're taught about this separately. This is to allow them to develop judgements about whether particular individuals are "fit" to order them about, and if humans give conflicting orders, which to obey.

The result is: "By the Three Laws, the human-beings-like-the-others are of lesser account and can neither be obeyed nor protected when that conflicts with the need of obedience to those like ourselves and of protection of those like ourselves". The story ends shortly after - having established the threat there is no need to describe the violent consequences.