Followed by a Gaya sraddham, having renounced Kothavarangai at Akshaya Vatam, can we partake any Kadamba prasadam at Mutt or Ashramam that includes Kothavaranga? What must we do if we partake this not knowing of its inclusion?

Since Kothavarangai should not be mixed with tamarind, it will not be included in the Kadambam; hence the first part of the query is not applicable. If a vegetable that has been given up is included in any Bhagavad prasadam, we can remove it and continue to partake rest of the prasadam; or if one even not partake such a prasadam; it is as per their preference. Thought of Krishna alone is the atonement for partaking a food item that has been given up without being aware of its inclusion.