The 1936 Berlin Summer Olympics were supposed to be a grand display of aryan superiority under the Nazi government.
The 1978 football World Cup in Argentina was used by the military dictatorship as a smokescreen, in order to cover up their human rights violations and State-sanctioned terrorism. While citizens were being tortured and summarily executed in ilegal detention, centers such as the ESMA (Army's Mechanics School), literally blocks away, people cheered during matches at the Monumental Stadium in Buenos Aires.
What are some other instances in which major sports events were used by governments with political motives?
Arguably many of the modern Olympic games have been used for political purposes, mainly to showcase the host country and announce their presence on the global stage. While there are several examples to look at (Rome 1960, Tokyo 1964, and Munich 1972 all come to mind, seeing how all three were former Axis powers from the Second World War; the last one also had the distinction of the Munich Massacre, which was clearly a political event), I'm going to look at the 1980 Summer Olympics, held in Moscow, then of the Soviet Union.
It is now most famous for the US-led boycott (the second in a series of boycotts at the Summer Olympics. The 1976 Montreal games had most of Africa boycott due to the New Zealand rugby team touring apartheid South Africa just prior, but this has largely been overshadowed by the 1980 Moscow and 1984 Los Angeles boycotts, which were led by the US and USSR, respectively), but the 1980 Summer Olympics were meant to be a showcase of the wonders of socialism. It would demonstrate to the world that socialism clearly was better than capitalism, and what better way to do so than by having an Olympics game in the capital of the Soviet Union? At this point I'll note that Moscow had made its first bid for the Olympics in 1976, losing to Montreal, but was successful in its second attempt, defeating Los Angeles (who would be awarded the 1984 Summer Olympics).
The Soviet Union had been a latecomer to the Olympic movement, only joining in 1952. Until that time it had dismissed the Olympics as a bourgeois movement, and not worthy of socialist attention, instead starting their own quasi-Olympic games, the Spartakiad, only for Soviet athletes.
But in the post-Second World War era this view was softened, and after intense lobbying by leading sports figures, Stalin relented and allowed them to participate. This was done as it was decided they could showcase the advantages of socialism through athletes, and how much better they were. Indeed Stalin actually initially agreed to let the Soviets make their debut at the 1952 Winter Olympics, but when his officials couldn't guarantee a gold-medal sweep in all events, he backed off, though they managed to change his mind by summer. It paid off too, as the Soviets won 71 medals (22 gold), finishing just behind the US in total medals (76), though the Americans had the clear advantage in golds (40). This started a trend where the US and USSR fought for medal supremacy, with each side dominating the medal count over the following decades.
With this in mind it was determined in the 1970s that the USSR should finally host a games. This was agreed upon by the IOC, who awarded Moscow the 1980 games in 1974. A lot of money was spent to clean up Moscow and build stadiums and other assorted venues for the Olympics, and a massive training regime was put in place to ensure that it was clear who the top sporting nation was.
Of course the Soviet invasion of Afghanistan in December 1979 and the subsequent US-led boycott meant that nearly every state not aligned with the Soviet Union skipped out. This only fueled the goals behind hosting, as the USSR trounced their allies in terms of medals, with 80 gold and 195 total medals (East Germany was a distant second, with 47 gold and 126 total). It allowed the Soviet authorities to show, to their domestic audience at least, how great their system was, and thus was partially able to fulfill the goals for hosting the games.