PC Ian Bell, 46, had admitted sending the messages while off-duty and drunk. They include one which said 'You like sex?' and another which read 'Please delete everything'.
PC Ian Bell called teenage girl 'cheeky' in messages sent after his wife went to bedHe had accepted his actions amounted to misconduct but denied gross misconduct, a sackable offence.
The girl was living in a children's home and married man PC Bell had been investigating reports that another young resident had gone missing, a two-day misconduct hearing in Wakefield was told.
He visited the home on July 19 last year and had a conversation with the girl, who cannot be named.
In the early hours of the next day, he began sending her messages via the social media application Snapchat which had "no legitimate policing purpose", Claire Watson, legal representative of the force's Professional Standards Directorate, said.
She said: "He knew, or ought to have known, that his actions were an abuse of his position as a serving police officer and was likely to cause distress and anxiety to the young person to whom he was sending these messages."
At first, the girl told the children's home staff she thought she had a "Snapchat stalker" and did not know who was sending the messages, the hearing was told.
Ms Watson said that at about 1am the girl then asked a member of staff: "If it was a police officer who sent the messages, what would happen if he could come and find me?"
She said: "The messages included references to seeing [the girl] in a towel, a question about whether she was taking drugs at parties and whether she likes sex."
Adam Birkby, representing PC Bell, had said: "This is a case of drunken idiocy and not a sophisticated attempt to exploit a vulnerable person for sexual purposes."
PC Bell said he had messaged the girl to thank her for her help in the missing persons investigation and had also wanted to offer her advice.
He had been concerned the girl could be taken advantage of but accepted that this had been an inappropriate way of acting on his concerns.
He said was not acting out of a "sexual motive" and shut down the conversation because he was "panicked" at how the messages could be misconstrued.
He said: "I was off-duty, intoxicated, talking to a 14-year-old person. I let the force down and I let myself down."
PC Bell has given 23 years of service to West Yorkshire Police and was based in the Wakefield district.