Coote, 43, was sacked last year by PGMOL, the English football referees' body, after a video circulated on social media of him making derogatory comments about former Liverpool manager Juergen Klopp.
That incident sparked further investigations, which led to Coote's electronic devices being seized by police in February 2025, prosecutor Jeremy Janes told Nottingham Crown Court.
Janes said Coote had, in January 2020, downloaded a single Category A video, the most serious kind, which showed a 15-year-old boy dressed in a school uniform performing a sex act.
The charge of making an indecent image relates to downloading, sharing and saving material, and Janes said there was no suggestion Coote was involved in the making of the video.
Coote pleaded not guilty at his first court appearance in September, but changed his plea to guilty the following month.
His lawyer, Laura Jane Miller, said Coote had been taking cocaine while struggling with his mental health at the time of the offence and was "deeply ashamed."
"He has lost his career, and he has lost his reputation, and those have been lost in the midst of... intense media scrutiny," she said.
Judge Nirmal Shant, however, said: "Some might say he has brought it on himself."
The judge told Coote: "You have had a spectacular fall from grace."
Shant imposed a nine-month prison sentence, suspended for two years, and ordered Coote to carry out 150 hours of unpaid work.