He was Lord Mayor of York in 1663, and was made a deputy lieutenant of the West Riding of Yorkshire in 1665.
In 1668, Henry moved from York to a new country estate at Escrick, where he had been buying land for the past several years.
He moved once more before his death, to another estate at Long Marston, leaving Escrick to his eldest son.
By his third wife, Susannah Lovell (d. 1701), he had several children, including Edward Thompson (1670–1734), who would inherit Long Marston.
One of the previous incumbents, Sir Thomas Osborne, had tried to stop Henry from taking his seat after he thought his son had been assured to be the favoured candidate.