The agency was handed the retained brief following a competitive pitch process with the software firm, which is headquartered in California and has nearly 15,000 global staff.
MHP has been given the task of raising Adobe's corporate profile, as well as driving greater understanding of the company's product portfolio and providing lobbying support where necessary.
The outgoing agency is understood to be Hume Brophy - it had listed Adobe as a client on the Association of Professional Political Consultants' register between 2012 and the most recent register, which covers the three months to 31 August 2016.
The APPC also reveals clients of MHP, which, according to the PRWeek Global Agency Business Report 2016 is the UK's 20th largest PR firm, and the world's 60th - to include Arla Foods, Boots Opticians, Crossrail and Wolseley.
Adobe and MHP are yet to respond to PRWeek's request for comment.
Adobe has worked with Edelman in the US for more than a decade.