We met her in Conscience Of The King. Her scientist husband Dr. Tom Leighton alerted Kirk to the fact that actor Anton Karidian might really be "Kodos the executioner". He was found dead soon thereafter.
Mrs. Leighton asked Kirk, "Why was Tom killed, if what he suspected [about Karidian being Kodos] wasn't true?"
"I don't know", Kirk replied, "I'll try and find out."
"Oh, Jim! How I'll miss him!"
Kirk hugged her for a split-second before she ran off to commence grieving for her poor murdered husband.
She called Kirk "Jim" because he is a family friend. This woman was never Kirk's girlfriend. She was Tom Leighton's wife/widow. The only reason Kirk hugged her is because she was recently widowed, and he was consoling her. Kirk never had a sexual relationship with her, just like he never had a sexual relationship with the vast majority of the women on TOS.
She was a perfectly respectable married lady, not some alien hoochie-coochie girl, but on Star Trek that was normal. We met perfectly respectable women like Mrs. Leighton on Star Trek all the time.
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