Gap between the gum and the crown can be due to multiple reasons which can range from poor oral hygiene to an underlying periodontal infection. The first and foremost thing you need to do is maintain a good hygiene and even if you do make sure you use a mouthwash/warm saline gargles twice daily. Secondly, visit your dentist as soon as possible to rule out any underlying issue. Thirdly, avoid picking with a toothpick and harsh brushing.
As for symptomatic relief, you can use Stolin gum paint on your gums.
Next Steps
Maintain oral hygiene.
Visit a dental clinic
Use stolin gum paint
Health Tips
Do not use toothpick
Do not brush your teeth harshly.