Jenison, who came of an old family of Newcastle merchants, was returned for Northumberland on petition in 1724. He usually supported the Government, but was absent from the division on the excise bill in 1733 and voted for the repeal of the Septennial Act in 1734. Re-elected in 1734 after an expensive contest, he was made master of the buckhounds in succession to Lord Tankerville, with whom he was politically connected, in 1737. Probably owing to the cost of the contest of 1734, which seriously impaired his fortune, he did not stand for Northumberland in 1741, when a promise of a seat obtained for him by Tankerville from Walpole in 1740 came to nothing.
He died 15 May 1758.
