Gillingham, launching speculation that he wanted a move away from the club. Although he subsequently apologised for his refusal to play, he did not play again for Wolves and negotiations were later entered into with Premier League team Crystal Palace. Crystal Palace On 31 January 2014 Hennessey joined Crystal Palace on a three-and-a-half year deal for an undisclosed fee, with media speculation suggesting it to be an initial £1.6 million, with add-on clauses. Hennessey made his debut for the Eagles in the team's final match of the Premier League season, a 2–2 draw with Fulham on 11 May 2014. Hennessey's first appearance of the 2014–15 season came with a 3–0 win at Walsall in the Football League Cup second round. On 25 October he made his first appearance of the season Premier League as a substitute for injured first-choice goalkeeper Julián Speroni in a 2–2 draw at West Bromwich Albion. He ended the season starting in Palace's final two league fixtures, wins against Liverpool and Swansea City, keeping his first Premier League clean sheet for the club in the latter match. After starting in Palace's 2015–16 Football League Cup second and third round matches, Hennessey was selected to start the Premier League fixture against Watford in place of Alex McCarthy. He kept a clean sheet as Palace won 1–0 at Vicarage Road and retained his place as the team's first-choice goalkeeper, making five consecutive league appearances. He played in the 2016 FA Cup Final, lost 2–1 to Manchester United after extra time. Hennessey signed a three-year contract in July 2018. However, in the ensuing season, he competed with new signing Vicente Guaita to be the starting goalkeeper. The Spaniard was first choice in the following season. Hennessey was sidelined from October 2020 until the following February, with a thigh injury from an international match. He left Crystal Palace in July 2021 upon being released. Burnley On 20 July 2021, Hennessey joined Burnley on a free transfer following his release from Crystal Palace, signing a two-year deal. He made his debut on 25 August in the second round of the EFL Cup away to Newcastle United, keeping a clean sheet in a goalless draw and then saving from Allan Saint-Maximin and Miguel Almirón in a 4–3 penalty shootout win. In the league, he was back-up to Nick Pope and played only in two 3–1 losses at Manchester United and Leeds United