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The phrase "a small influence on" is correct and usable in written English.
It can be used to describe a minor effect or impact that something has on another thing.
Example: "The new policy had a small influence on employee productivity, but the overall results were still positive."
Alternatives: "a slight impact on" or "a minor effect on".
Exact(60)
Overall, block duration had a small influence on task choice frequencies (Fig. 4c).
Relative availability of oxygen and hydrocarbons exerts a small influence on results.
The virtual mass force has a small influence on the investigated bubbly flow characteristics.
Treatment design tended to have a small influence on effect size.
The players win the games and managers only have a small influence on the impact the players have.
Remarkably, the composition of the coating has only a small influence on the hardness and modulus.
The temperature, however, has a small influence on the phenols adsorption on Zn-Elut.
Where you live and the cost of living there has only a small influence on that number, he added.
Moreover, a variation of the cutting and feed speeds has only a small influence on the PCBN insert quality.
Instead, researchers think, many hundreds or thousands of genes could be involved, each with a small influence on a person's overall intelligence.
The presence of chromium has a small influence on stacking fault energies on planes perpendicular to the basal plane.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com