After a two-year spell in Trenčín, he returned to Žilina, where he was the second-choice keeper, the first-choice being Ján Mucha. Despite not playing at all, he was linked by the English Premier League side West Ham, where he was loaned for 6 months. In January 2006, he returned to Žilina, and he established himself as the first-choice keeper. He won Corgoň Liga with Žilina in 2006–07 season.
On 26 June 2008, he signed a three-year contract with Romanian side FC Vaslui. He was bought to replace Cristian Hăisan, while he was suspended, but Kuciak established himself quickly as the first-choice keeper. He was the only player that season, who played every minute, and the first in Vaslui's history who achieved this feat. At the end of the season, he established another record, having the most clean-sheets in Vaslui's short spell in Liga I. Also, he captained the team a few matches, while Cânu was on the bench. On 4 August, he received a red card in the Europa League match against Omonia Nicosia. On 26 September, after a conflict with team's coach, he was relegated on the bench, and seemed to lost his place in the team, but because of Hăisan's poor rating, re-established as the first-choice keeper. At the end of the season, he established another record for Vaslui, saving 4 out of 5 penalties, and also receiving only 17 goals in 23 matches. After three seasons at Vaslui, Kuciak left the club in the summer of 2011.
Kuciak joined the Polish club Legia Warsaw in August 2011. In August 2013, he extended his contract at the club until 2016. West Ham United In January 2013, West Ham United were reported to be in talks with the goalkeeper.
On 1 February 2016, Kuciak signed an 18-month contract with Hull City. Kuciak made his debut on 23 August 2016 in the second round of the EFL Cup away to Exeter City that City won 3–1.
International career
He made his international debut for the Slovakia national team in a 2–1 away win against UAE on 10 December 2006. After a two-year absence in the national team, he was called again in 2009, because of his impressive form at Vaslui. He was named in the provisional 30-man squad for 2010 World Cup. He played in the Slovakia's second warm-up game, where he was replaced in the 85th minute, because of an injury. However, he was kept for the final 23-man squad, for 2010 World Cup, where he wore the Number 23 Shirt.
