The following page of the guide to Syberia: The World Before explains what secondary objectives are and when you need to complete them.
In Syberia: The World Before you will have to complete all main objectives of the current chapter one by one to push the storyline forward. You can find all current objectives by clicking on the exclamation point located in the upper right corner of the screen.
Every now and then you will come across optional side objectives, which you always need to do before the main objective, because by pushing the plot forward you can leave an area or even finish the current chapter. Occasionally, the secondary objectives will include detailed instructions about when you need to complete them.
The hints available in the game unfortunately never refer to secondary objectives. You will have to look around carefully on your own and find the appropriate places or objects.
You can find all the main and secondary goals, both ongoing and completed, by clicking on the book icon at the top right of the screen. The first tab of the book contains the objectives, divided by color. Optional secondary objectives are written on yellow pages.
You don't have to do all the side objectives to complete the game - they only let you get to know the characters and the story better. However, completing the side objectives is necessary to unlock several trophies.