He started his career with the Johannesburg-based Golden Lions. He represented them at Under-16 level in 2007 and at the 2008 Craven Week tournament. After playing for the Golden Lions U19 side in 2008 and 2009, he made his first class debut for the Lions in their 2010 Super 14 match against the Stormers. In just his second match, he scored a hat-trick in the Lions' 65–72 defeat to New Zealand side the Chiefs. He eventually made a total of ten appearances during the tournament. He played for the Golden Lions U21s in the 2010 Under-21 Provincial Championship Group A, making three appearances. He was named on the bench for the Golden Lions' 2010 Currie Cup Premier Division match against the Pumas, but failed to make an appearance.
The following season, he made four appearances for the Golden Lions in the 2011 Vodacom Cup and eleven appearances for the Golden Lions in the 2011 Under-21 Provincial Championship.
In 2012, Mjekevu returned to his hometown of Durban to join the Sharks. He made six appearances for them during the 2012 Vodacom Cup, scoring one try in their match against a Free State XV. He played in 13 matches for the Sharks XV during the 2012 Under-21 Provincial Championship.
However, he failed to break into their Currie Cup squad and the following season, he played in the 2013 SARU Community Cup for Durban-based club side College Rovers. He scored a brace of tries in their match against SK Walmers and a hat-trick the following week against Villagers Worcester as College Rovers finished runners-up in the inaugural competition.
In 2013, Mjekevu signed a deal to join French Top 14 side Perpignan in 2013. He scored four tries in 15 appearances for them in the 2013–14 Top 14 season, which wasn't enough to prevent his side finishing 13th and being relegated to the Rugby Pro D2. He also scored two tries in three appearances in the 2013–14 Heineken Cup.
He scored ten tries in 18 appearances in the 2014–15 Rugby Pro D2 as Perpignan finished in third place on the log, before losing to Agen on number of tries scored in the promotion play-offs.
After the 2014–2015 European season, Mjekevu returned to the Sharks prior to the 2015 Currie Cup Premier Division, signing a two-year deal with the team.
Mjekevu was named in the Sharks squad for the 2016 Super Rugby season, but failed to make any appearances in the first thirteen rounds of the competition and was loaned out to the Port Elizabeth-based Kings for the remainder of the season.
Mjekevu earned a number of call-ups to junior South African sides. He was selected in a South African Under-18 Elite Squad in 2008 and subsequently selected in the final South African Schools squad in the same year. In 2009, he was included in a South African Under-18 High Performance Squad and he was a member of the South Africa Under-20 team that played in the 2010 and 2011 IRB Junior World Championships.