He was a member of the I and II Riigikogu from 1920 to 1926. From 18 November 1921 to 20 November 1922, he was also Speaker of the Riigikogu.[2] He resigned as Speaker upon being appointed State Elder, an office he held until 2 August 1923. After Kukk left the Riigikogu, he was succeeded by Johann Kana.
Kukk was an active promoter of the cooperative movement, and also served as chairman of the council of the Estonian Cooperative Union, Director of the Central Union of the Estonian Consumers Union, and for some time also as chairman of the board of the latter.
Aside from his career in politics, Kukk also worked as a banker. In 1920, he became chairman of the board of the Estonian People's Bank, and he served as director of the Bank of Estonia from 1924 to 1926.