I’m getting paid 4k (5k if I were to add in bonuses) currently and I’ve had 6 years of experience in the media industry. I feel I’m slightly underpaid, is it possible to ask for 6k in my next job?

If you can support with results of your past work, this refers to any significant project experience and prove how you've handled complex situations well then I see no reason why not.

When hiring, the organisation will have a budget.

Write your current & expected salary in your resume? If within their budget and your skill matches. They will call you for an interview.​​​