This sounds like "The Partner" (also translated as "The Companion") by Ilya Varshavsky, a Soviet science fiction writer. The story was originally published in Russian in the 1960s.

The plot matches your description very closely:

• A lonely woman purchases a male android companion
• She keeps sending him back to the factory for upgrades to make him more human-like
• First he's too basic and mechanical
• Then he lacks opinions of his own
• Then she realizes his feelings are programmed, not genuine
• Finally, she requests they give him free will, despite warnings that he might choose not to return to her
• The story ends ambiguously with her waiting, not knowing if he'll come back

Ilya Varshavsky (1908-1974) was a prominent Soviet science fiction writer who often explored themes of artificial intelligence, human nature, and the relationship between humans and machines. His stories were frequently included in Soviet science fiction anthologies that were translated into English during the 1960s-1980s.

The story explores philosophical questions about what makes love and companionship authentic, and whether programmed emotions can ever be considered "real." It's a poignant examination of loneliness and the human desire for genuine connection.