[10] Although he began his writing career with short stories, Abbensetts had been attracted to playwriting after seeing a performance of John Osborne's Look Back In Anger in Montreal, while studying at university there, thereafter deciding to move to London to become a writer.
[5][11] Abbensetts's work in stage drama debuted in 1973 at the Royal Court Theatre Upstairs with his play Sweet Talk, which had a cast including Mona Hammond and Don Warrington, and was directed by Stephen Frears.
[12] Abbensetts became the Royal Court's resident dramatist, and Sweet Talk won the George Devine Award,[13] shared with Mustafa Matura.
[13] With Abbensetts' health declining in his latter years as a result of Alzheimer's disease,[2][24] a tribute was organised for his benefit by Anton Phillips on Sunday, 9 December 2012: a rehearsed reading of Sweet Talk, directed by Phillips and attended by Abbensetts himself, was held at the Tricycle Theatre, with many well known figures in Black theatre and arts in the audience, including Yvonne Brewster, Don Warrington, Rudolph Walker, Oscar James, Allister Bain, and Errol Lloyd.
[25] Abbensetts died aged 78 on 24 November 2016, survived by his daughter, Justine, from his relationship with Anne Stewart, and by two grandchildren, Sean and Danielle, as well as a sister Elizabeth.
[26] A property that Abbensetts bought in 1990 in Kilburn, London, where he lived for 16 years and wrote the series Little Napoleons for Channel 4, was chosen as the location for a Nubian Jak Community Trust blue plaque honouring him.