Is mulching with compost needed if bark chips used?

Hi - I am a relatively new gardener - I created a garden last year, mainly shrubs and perrenial flowering plants, and covered it with bark chips for weed suppression and aesthetics and I am now learning how to maintain it.  Last Autumn and now coming up for this Spring, I have read several articles recommending putting a mulch of compost across the garden to improve soil texture and to add nutrients.  The only way I can think of to do this would be to take up all of the bark chips first - which would be a very onerous task!  Is this actually necessary if the garden is covered in bark chips anyway?  Any help gratefully received!  Thanks