Welcome to NGA, bunnie22. I assume you are describing a hydrangea macrophylla.
Will it bloom again? Yes but we need to know the name of this variety to know when. It can bloom again either in Spring 2021 or this summer.
Rebloomer or not? If you bought your macrophylla at a plant nursery, check to see if the plant label says whether it is a rebloomer or not. Hydrangeas bought at grocery stores or florists typically have hardly any information on their tiny plant labels. If that is where you bought yours, I would assume it is not a rebloomer.
Why are we talking about rebloomer hydrangea macrophyllas? A non-rebloomer macrophylla will only develop invisible flower buds once a year. Exactly when depends on how far south or north one lives. For me, the buds develop early to mid July. In NC, they may develop around late July or so. And so on. Those late Summer buds will open in the Spring and resemble small broccoli heads as they open. On the other hand, a rebloomer macrophylla will develop invisible flower buds as described above and then again: new stems that grow in Spring 2020 should develop invisible flower buds that will quickly open in early Summer or thereabouts.
Note: be aware that your plant was awakened at an unnatural early time in order to produce those pink blooms way early. So this year only, those time estimates that I mentioned may be off.
How far down to cut? The blooms attach to the stem with a little "string" called a pedicel. You can cut the bloom off right there if you want.