The 48-year-old former footballing star, known as Gazza, sent abusive tweets, text messages and telephone calls to Amanda Thomas over a period of 13 days in March this year.
He had a relationship with mum-of-two Mrs Thomas, 44, after meeting her in Boscombe, Dorset, where he underwent rehab treatment for his well-documented alcohol addiction.
Gascoigne, of Lilliput, Poole, also entered a guilty plea for assault by beating of the photographer Andy Stone, damaging his glasses worth £200.
On March 16 Gascoigne telephoned Miss Thomas at a Post Office shop where she works in Bournemouth, Dorset.
Mr Turner said that Miss Thomas told the defendant that she did not want to be contacted again but Gascoigne went on to send her text messages on eight occasions in the following days.
He added that on March 20, Gascoigne sent a “series of abusive tweets” including one post saying: “I am going to f*** him over”.
Mr Turner also read out insulting comments made by Gascoigne about Miss Thomas.
Mr Turner said: “As a result of those tweets, it encouraged other persons using Twitter which led to further harassment of Miss Thomas.
“Miss Thomas is at home with Mr Stone when she hears about the use of Twitter, at first she didn’t want to look at Twitter but she did.
“She was alarmed and distressed at their content.
“Mr Stone telephoned Mr Gascoigne asking him to stop, however the abusive tweets continued.”
Mr Turner said that the next day Gascoigne called Miss Thomas and told her “I am going to destroy you, I am going to hammer you, I am going to come into the Post Office tomorrow.”
He said that Gascoigne turned up at her place of work “causing a scene” and started shouting her name.
He then sent a further tweet saying: “If they do not sack the cow I am going to sue the Post Office.”
Gascoigne was later arrested after Miss Thomas made a complaint to the police.
Mr Turner said: “Miss Thomas says that her biggest concern was about losing her job and she did feel harassed by this course of conduct.”
Mr Turner said that the assault against photographer Steven Shepherd, who works with Mr Stone, happened as Gascoigne had been signing autographs and having his picture taken with fans as he sat at a cafe in Bournemouth town centre on June 16.
He confronted the photographer and demanded he hand over his camera’s memory card, causing him a one-inch graze to his head and damage to his prescription glasses in the process, Mr Turner said.
He added: “Mr Gascoigne becomes aware of the two photographers, Mr Gascoigne pursues Mr Shepherd from the cafe into Bournemouth Gardens and CCTV pictures show Mr Gascoigne running across the pedestrian precinct, reaching Mr Shepherd 40 metres away from the cafe.
“Mr Shepherd says after he took some photos, he was walking away and suddenly something struck him on the head and somebody grabbed his glasses from him.”
Mr Turner said that the victim said that Gascoigne said to “give him the f****** pictures” and Mr Shepherd handed over two memory cards and he went on to report the incident to the police.
Mr Turner said that Gascoigne’s basis for pleading guilty to the two charges relating to Mr Shepherd was that he was reckless and had not intended to hurt him or cause damage to the glasses.
Former Newcastle, Tottenham, Lazio and Rangers midfielder Gazza moved to Bournemouth in 2012 to attend the Providence rehab clinic.
He lived in a penthouse flat overlooking the sea in the the Boscombe area to begin with before moving into a harbour-side flat in Poole.
He now lives in a rented bungalow in Poole.