The phrase "huge sigh of relief" means a very big and noticeable feeling of being relieved or feeling less stressed about something that was worrying you. Full definition
And it's the reason that researchers dependent on federal money, and especially those in the physical sciences, are heaving huge sighs of relief over a disaster narrowly averted. (sciencemag.org)
Nathan Redmond led a lightning break and danced past Zanka to square for Austin, and as an open goal gaped at his mercy, the opposite number 10 fired the wrong side of the post - huge sigh of relief by all involved with Huddersfield Town. (htafc.com)
While these are just early results, this could bring either a huge sigh of relief to Europe — or another disastrous deadlock. (businessinsider.com)