First, -い versions are i-adjectives but -やか versions are na-adjectives. Second, I feel -やか versions have slightly different primary meanings.
- 緩い【ゆるい】: loose, easy
緩く引っ張る: to pull weakly
- 緩やか【ゆるやか】: gradual, slow
緩やかに引っ張る: to pull slowly
- 軽い【かるい】: light (weight), light (mood), casual, easy, shallow (mind)
軽く走る: to run for a short time, to run casually
- 軽やか【かろやか】: light (footstep), cheerful
軽やかに走る: to run cheerfully, to run with light steps
I chose these examples to illustrate the difference, but there are times when they are interchangeable. 涼やか【すずやか】 is rare and I'm not sure how it's used.
I think やか is just another suffix used to make an adjective, and I doubt it has its own noteworthy meaning. It's no longer productive (i.e., new words using -やか are unlikely to be coined any more). As Earthliŋ mentioned, many (most?) -やか adjectives do not have their -い counterparts at least in modern Japanese (e.g., はなやか, みやびやか, ひそやか, しとやか).
When you encounter both -い and -やか versions, I think you should just remember them as different words. Just as you should distinguish numeric and numerous, or temporary and temporal, as two different words with different meanings.