The Information Commissioner's Office has decided the three organisations have fallen short of its guidelines that organisations should respond within 20 working days of receiving FOI requests. It is monitoring their performance for three months until 31 March.
Commenting on the ICO’s concerns, the information commissioner Christopher Graham said: "Responding to FOI requests within the statutory time limit of 20 working days is basic to delivering transparent and open government.
"While extensions may sometimes be justifiable for particularly complex requests, these three authorities have been selected for monitoring after serious shortcomings were identified in the time each of them has been taking to respond to FOI requests."
It is the second time in three years that the Cabinet Office has been subject to FOI monitoring by the ICO.
Defending the Cabinet Office, a spokesman blamed a high level of requests around the funeral of Margaret Thatcher, among others.
"There was only one quarter in 2013 where the Cabinet Office’s performance was narrowly below the level the commissioner previously found acceptable.
"During this quarter we had a 23 per cent increase in requests, most of which focused around the funeral of Margaret Thatcher, the Leveson Inquiry and information about Jimmy Savile. Even then we still answered more than eight out of ten requests on time."
The CPS admitted to a strain on resources, claiming the number of FOI enquiries it has received has increased by 4 per cent since 2011.
A CPS spokesperson said: "We have recently recruited further staff to address the increase in workload and are already seeing improvements in the timeliness of our responses during this monitoring period."
Hackney Council said it carried out a review of its response to FOI requests in November 2013 and recently introduced new measures to process requests more efficiently.