Hardie on formica is not a great formula for longevity...good luck! Assuming you used screws to anchor the Hardie, and most formica counters are installed over particle board, you opened up a path for moisture to swell up the particle board a certain path for failure in the long run. Pay particular attention to the area around the sink to seal the edges before installing the sink and wipe up any spills immediately, and it can last awhile. Next time, tear the old counter off, build it up with plywood, then proceed to tile over a waterproof membrane or cbu (like Hardie).
Consider checking on the sink manufacturer's website...I know at least some of them offer longer screws/clamps for thick counters. Also, you might call a granite fabricator, they may have some you could buy. I wouldn't rely on filling the gap with caulk...I think in the long run, it would look funky.