Connelly was a member of the Centre-Right faction of the Liberal Party. He served as a member of several parliamentary committees including Defence, Foreign Affairs and Trade; Infrastructure; Transport and Cities; National Broadband Network (NBN); Appropriations and Administration; and Mental Health and Suicide Prevention. On 16 September 2021, the Prime Minister confirmed the Federal Government would work with the WA government to invest in a large dry dock at Henderson. On 19 March 2021, the Australian Electoral Commission published a report proposing a redistribution of WA's federal electoral divisions under which the Division of Stirling would be abolished. The redistribution was confirmed on 4 June 2021. With the Division of Stirling set to be abolished at the
2022 Australian federal election, Connelly announced that he would challenge incumbent
Moore MP
Ian Goodenough for
preselection for that seat. This was against the wishes of senior party figures, including Prime Minister
Scott Morrison, who wanted Connelly to run for the
Division of Cowan. Moore is a safe seat for the Liberal party, with a margin of 11.6% in the 2019 election, whereas Cowan is a marginal seat, with Labor winning it by 0.8%. Connelly said "(Moore) is where our kids play soccer, where my wife and I participate in our local RSL, it's where we surf, shop and visit friends." In a statement on 4 September 2021, Connelly confirmed he would nominate for the federal seat of Moore. Connelly said "Electoral boundaries may have moved, but my local community has not." Connelly narrowly lost being preselected for Moore 39–36 after preselectors voted at a meeting on 22 September. He was later endorsed as the Liberal candidate for
Cowan, where he was defeated by incumbent MP
Anne Aly. In January 2024, Connelly nominated himself for Liberal preselection for the Division of Moore again, going up against Ian Goodenough again, this time succeeding on 17 February. He was defeated by the
ALP's
Tom French with a swing of 4.9%. ==Personal life==