Sparkpost blockem my email in the form of firstname.lastname at gmail.com.
I am getting:
554 5.7.1 recipient address suppressed due to customer policy
Why would sparkpost block private email addresses like that?
This means that the address is in your suppression list. Addresses get added automatically based on hard bounces, unsubscribes, and spam complaints.
SparkPost are entirely open to questions. You can contact support, or sign up to the community Slack and ask there. (You don't even need to ask because if you do a search in Slack you'll see this question has been answered many times.)
Also, you can read the support documentation about Suppression Lists, and look at the Suppression List API docs.