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Reading the name like this "Arh- ni - kha" I pronounce it exactly as you want it pronounced :)
Reading it like Anika, i pronounce it how everyone else is pronouncing it.
So all i think you need to do is change the spelling :D
No need to give up a name you love :D
What about Chardonnay, Serenity, Tia, Ocean (OH SHEE ARN), Blessing, Sapphire, Briella, Satarrah.
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I know an 'Arnica' which is the spelt the way that I think u like to pronounce it.
It is also a medication for sore muscles.
The Anika I know pronounces her name A - NEE - KA and everyone she meets always pronounces it that way.