"exercise an influence" is correct and usable in written English.
You can use it when referring to the power or effect of a person or thing on another. For example, you might say, "Her dedication to her craft has continued to exercise an influence on her peers."
It allows Jews to exercise an influence out of all proportion to their mere 2% of the population.
These set a fashion in church building and decoration that was to exercise an influence for many centuries.
That means, as it has for years, that people from, say, fifteen to twenty-five years of age exercise an influence on what gets made by the studios way out of proportion to their numbers in the population.
Where "political campaign contributions (financial or otherwise) exercise an influence on the nature of posting received," the desirability of a posting should correspond to the size of the campaign contribution.
Though its literary culture has no single figure whose influence can be compared to that of Italy's Dante or England's Shakespeare, successive periods have seen its writers and their language exercise an influence far beyond its borders.
Indeed, through their influence on the latitude of the jet stream and the wavelength (that is, the spacing of cold trough and warm ridge regions) in the upper westerlies, these ocean anomalies exercise an influence over the atmospheric circulation that spreads to all parts of the hemisphere.
General Smuts, the Prime Minister of South Africa remarked: "Mr Lloyd George once said to me that the Manchester Guardian exercised an influence entirely out of proportion to its circulation.
Beautiful and intelligent, she exercised an influence over her husband that was decisive in establishing and maintaining Sir Robert Walpole as prime minister (1730 42).
Other commentators, however, while sympathetic to Celyn's family, are less than enchanted with the sight of the mummy-warriors marching forth yet again, exercising an influence they believe is disproportionate to their numbers.
In 1949, when the People's Republic of China was formed, Christians represented only 1 percent of the Chinese population, but they exercised an influence out of all proportion to their size.
And carnivores? "Because large carnivores regulate other predators and prey, exercising an influence on the ecosystem far out of proportion to their numbers, their protection and reintroduction is crucial".
