Ramayana, the greatest epic as you rightly said, has a number of characters.
The character of Mata Kaikayi stands to be the most important one as the story of Ramayana has revolved around her cunningness.
Even though a majority considers Mata Kaikayi as the reason of Sita Ram leaving for vanvas along with Lakshman, and it is true also, there's no doubt in that, still Manthra is the actual reason why Mata Kaikayi even made a demand like that.
Manthra was just another sevak at the Rajmahal of Raja dashrath who was very close to Mata Kaikayi.
When Maharaj Dashrath decided to coronate Shri Ram as the next King of Ayodhya, Mata Kaikayi was very happy.
She ecstatically ran around the entire kingdom helping in the preparations of the coronation but as soon as she gave the good news to manthra, manthra played her part.
She manipulated Mata Kaikayi by saying that Raja dashrath has always favoured Ram more than your son Bharat and that Bharat is more deserving of the crown.
Mata Kaikayi did try to hold back but when the question of her son comes, a mother is not capable of distinguishing between what is right and what is wrong.
Mata Kaikayi fell into the clutches of Manthra's evil mind and decided to get Bharat coronated in place of Shri Ram.
Mata Kaikayi had once been given two boons by Raja Dashrath, one of them she had already availed by asking Raja Dashrath as her husband, the other one she had said that she would ask for some other time.
When the time came, Mata Kaikayi rushed to Raja Dashrath and asked her to send Ram to the forest for 14 years and coronate Bharat in place of him as her second vardaan.
Raja Dashrath wouldn't have conceeded to the woman's demand but
Raghukul reeti sada chali aayi
Praan jaayi par vachan na jaayi
He was a man of his word and so had to send bhagwan Ram to the forest as vanvaas for 14 years.
Mata Sita and Lakshman after hearing this requested Lord Ram to take them along with him and that is how Sita Ram Lakshman sahit had to leave for the forest.
Mata Kaikayi, very happy by this, went to Bharat and told him about the incident. She thought that her son would be happy if he gets to become the king but instead he despised his mother for her act and left home saying he won't come back until his brother with Mata Sita and Lakshman comes back to dwell in Ayodhya.
It was then that mata Kaikayi realised what she'd done was wrong and hated Manthra for getting into her head.
This is how Manthra is an important character when talking about her contribution.
The rest is Ramayana!
I hope the answer was sufficient to clear your doubt, if there is something else, you can post or check my previous answers on Ramayana and Mahabharata for more information.
Siya-var Ramchandra ki Jai! ❤️
Please chant
Hare Krishna Hare Krishna
Krishna Krishna Hare Hare
Hare Ram Hare Ram
Ram Ram Hare Hare.