It affects your health greatly.
It has a great effect on your health.
With the verb influence, you could use any of those adverbs, but don't use on- use that with the noun (have an influence on). However, influence doesn't work for me with health- have an effect/impact, etc, work better.
(I guess the idea of 'on' occurred to you because it could be used after and adjective: 'has a strong/great/serious/deleterious... effect/influence on your health'.)
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