How do you show that a character is suffering from lovesickness in a story plot, but not distract the reader from the actual main plot of the story?
Consequences are key. Show the consequences of being lovesick in a way that will relate to the reader. Describe the character as losing sleep, not eating, etc. Pick a tone that you want to stick with though. Is the lovesickness going to be a good thing in the overall scheme of the story, or is it going to be negative? Answer that, and then branch off from it. If it is a good thing, make sure to follow every symptom with a positive spin from love story.
The secret is to make your scenes perform more than one task at a time.
You don't need to dedicate whole scenes/chapters to showing your protagonist's lovesickness. That's not the story, it's not action, it's an emotional undertone. The reader will remember they're lovesick, trust them.
Concoct events that propel the story's momentum while occasionally reminding the protagonist of the pain of love lost. But don't overdo it. Trust your reader.
I feel like this question is two fold - you're asking about how to write a pining character, and about how to make it non distracting.
There's always someone who's going to be distracted by an element of a story, maybe because they really like it, maybe because they really hate it. I usually just try to spend less 'screen time' on things that I want to mention but I don't want readers to focus on. I also try not to make it too powerful. A random side character's crush could be a funny inside joke, but a random side character's deep attraction to their very best friend, revealed only after one of the pair is dead... that's going to stick in a reader's head a bit more.
One example is from the Heroes of Olympus series. Think of the way Nico's crush on Percy is treated in House of Hades (iirc) - it's major, a big deal, and the characters focus on it for a whole chapter. In Blood Of Olympus, it's mostly just a line or two being thrown around, an inside joke for Nico and Jason.
The more emotionally charged your writing about the crush is, the more distracting it'll be for readers.
Now, about writing the actual pining. Depending on how subtle you want it to be, you could have the character think about them all the time. Maybe if they see their crush's favourite thing, or smell their perfume. Maybe it's even more subtle, just comparing every person they meet to the crush, and finding that none of them match up. Their body language would also be different - more embarrassed around the crush, probably, but that depends on the character. Some people might go the "I'm pulling your hair because I like you" route. Tropes about behaviour from Japanese media like tsunderes and yanderes might come in handy as well. It's helpful to read a lot of (good) stories about crushes to see what else you can do.
You can have a few options here, each leading to a different result:
Muffle the lovesickness. The character's feeling would become a decoration to the plot, not the other way around. You would get what you are asking for at the expense of your character's development;
Make the reader feel the lovesickness with your character. Make this love front and center in your plot (just like your character feels it), and if some other events are important, they would have to wait for their turn;
Make the reader commiserate with your character's lovesickness. Sure love is the great thing, but hey, character, it's time to wake up and pay attention to what's really important!