sigh for (someone or something)
To long, yearn, or grieve for someone or something. He's been moping around the place sighing for his girlfriend, who just moved to Canada last week. I am happy in my life, I truly am, but I can't help sighing for some of the opportunities I passed up or squandered over the years.
sigh for someone
to release a deep breath, indicating anxiety about one's emotional attachment for someone. Dave spent a lot of his time sighing for Laura, on whom he had a crush. Laura has been sighing for some as-yet-unnamed young man.