This is the 12260th most frequent Russian word.
искреннее
Sincere; genuine.
The word "искреннее" here is used as an adjective to describe 'желание' (desire), illustrating an honest and true intention.
Он проявил искреннее желание помочь.
He showed a sincere desire to help.
In this sentence, "искреннею" acts as an adjective modifying 'благодарность' (gratitude), expressing an authentic and heartfelt emotion.
Мы чувствуем искреннюю благодарность за вашу поддержку.
We feel genuine gratitude for your support.
Here, "искренние" describes 'слова' (words), indicating that the words are expressed truthfully and without pretense.