Beginning in the first decade of the twentieth century, Jessup had poems published in a range of Australian publications, including regularly in Freeman's Journal from 1909. His focus was soon on black and white illustration and he contributed regularly to the Bulletin, Smith's Weekly, Aussie, The Comic Australian and other publications during the first half of the twentieth century.

The Bulletin published some 400 of Jessup's drawings, beginning with the 4 February 1909 edition, as well as articles and verse (The Bulletin v77#3974 11/4/1956). From 1913, he provided political cartoons for the Sydney Sun and was regularly published in The Land newspaper from 1917. During the early twenties, he regularly drew sporting portraits for the Sydney Evening News.

In the early 1920s, Jessup and Syd Nicholls provided comics for the 'Children's Budget' centre-fold in Woman's Budget, introduced on 13 November 1923.

At this time, the Sydney Sun trialled some comic strips, including Jessup's 'The Man Who Waited'. From January 1923 (when the Sun introduced a comic section) Jessup created 'Marmaduke', a fantasy about a young boy and his eccentric family and associates in the Australian outback. On 1 August, Jessup provided a full page cover illustration announcing the final episode of Marmaduke with a caricature of himself and many characters from the strip.

Jessup was a pioneer of Australian comics, writing and drawing for Frank Johnson Publications during the first half of the forties. These included 'Professor Darwin', 'Captain Spadger', 'Mulga Mick', 'Barbara' 'Pete' and 'Molly and Gran Pop'.

From 1941 to 1942, Jessup drew a regular cartoon 'Smith's Bush Naturalist...' for Smith's Weekly, providing fantastic combinations of animals and technology. He provided numerous illustrations and cartoons for the journal during the forties.

Jessup also wrote and illustrated several children's books. He painted in oils and watercolour, and was a member of the Royal Art Society and the Sydney Art Society. He was a founding member of the Society of Black and White Artists (now the ACA).