Mwesigwa began his career in 2002 with Ugandan side Villa SC, before moving to Icelandic club ÍBV in 2006.
After spending the 2010 season with Chinese side Chongqing Lifan, in February 2011 he signed a two-year contract with Kazakhstan Premier League side FC Ordabasy.
In February 2013, Mwesigwa extended his contract with Ordabasy, after winning the Kazakhstan Super Cup with them in 2012. Mwesigwa left Ordabasy in December 2014.
In May 2015, Mwesigwa was reported to have signed a three-year contract with Bidvest Wits of the Premier Soccer League, but contract negotiations broke down and Mwesigwa did not sign for Bidvest Wits.
After leaving Yenicami Ağdelen, Mwesigwa went on trial with V.League 1 side Hà Nội, which in 2016 was moved to Ho Chi Minh City and renamed Sài Gòn F.C. He spent the 2016 season with Sài Gòn, making 23 league appearances.
In January 2017, Mwesigwa went on trial with former club FC Ordabasy.