Hugh D. Badcock studied at Balliol College, University of Oxford, and was awarded the degree Master of Arts (MA) in mathematics. He was an associate member of the Institution of Civil Engineers and a member of the Royal Sanitary Institute. As an amateur astronomer he was also a fellow of the Royal Astronomical Society. During the Anglo-Boer War (1899-1902) he came to the Transvaal with the Imperial Yeomanry (a volunteer cavalry force raised from the class of landowners) in 1900. In October that year he was appointed as an assistant engineer to the town council of Pretoria, but in July 1901 became an inspector of machinery in the Department of Mines, stationed in Germiston. He was awarded the Queen's South African Medal with four bars and in 1901 married Margaret I. Close of Craddock.
In May 1902 Badcock was appointed Town Engineer of Pretoria, succeeding August Karlson*. Two years later he produced his Report on the sewerage and sewage disposal of the municipality of Pretoria. He also prepared a proposal for a comprehensive drainage scheme for Pretoria. Both schemes were approved in 1906. Badcock resigned in 1908, but remained on as town engineer until his successor, F. Walton Jameson, took over in March 1909. In 1911 he published a paper on 'The storm-water drainage of Pretoria (abridged)' in the Minutes of Proceedings of the Institution of Civil Engineers.
Badcock joined the South African Association for the Advancement of Science in 1903, and served on its council during 1907/8. His interest in astronomy was directed mainly towards sunspots and related indicators of solar activity, and in 1907 he contributed a paper on 'Solar activity in 1904-1907' at the association's meeting in Durban (Report, 1907, p. 30). The paper provided counts of sunspots, sunspot groups, and faculae for each month of the observation period. During the next year he presented additional observations for the period May to September 1907 under the title 'Note on solar activity 1907' (Report, 1908, p. 36). He was no longer listed as a member of the association in 1911. In addition to his work on solar activity he published a brief theoretical paper on 'The brightness of Mercury' in The Observatory (1906). He was an examiner in mathematics for the University of the Cape of Good Hope in 1906/7.