In Chekhov's "The Lady with the Pet Dog," was the essence of the story that Dmitri tries to make a real connection with someone?
The essence of the story, I believe, is found in its irony. Dmitri Gurov doesn't try to have a real relationship with Anna; ironically, he falls in love with her late in his life and finds himself amazed that he does love her deeply. Gurov had spent his life as a charming womanizer, a serial adulterer who had used women while holding them in contempt as "the lower race." When he does fall in love, his life becomes fragmented. He goes on living as society expects, but his "real life" is hidden:
". . . everything that was important, interesting, essential, everything about which he was sincere and never deceived himself, everything that composed the kernel of his life, went on in secret, while everything that was false in him . . . was on the surface."
The essence of the story is suggested in the transformative power of love and its effect on the life of a cynical man who had lived without it.
Further irony is developed in the story in that once Dmitri finds love with Anna, he cannot live a fulfilling life. He and Anna cannot leave each other, but the constraints of their society determine that they never can live authentic lives together. Dmitri recognizes what he perceives to be a fundamental truth of human existence:
". . . the real, the only interesting life of every individual goes on as under cover of night, secretly. Every individual existence revolves around mystery . . . ."
The true essence of the story is this realization, Dmitri's very poignant insight.