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That's not a sketch that are two short excerpts. Please post a complete sketch, what you posted doesn't compile as it is.
And define what "finally reporting" means. From the moment the measures are taken until the first packet on the server was received you measure minutes? How do you measure the time? What should the server return if that request is sent to it?
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What is the best way to use PUT on the Yun?
By providing the option "-XPUT" to the curl call? I don't know if that's the best way but it's one that should work at least. If you provide more information about your project we may have additional hints for you.