The contract is clear in that the adjudicator can extend the period for providing further information under clause W1.3(3), but only if you and the employer agree to that extension. Otherwise you and the employer must provide any information you or they wish to give the adjudicator within four weeks.
However, it is important to understand that the adjudicator also has the power to instruct you to provide further information related to the dispute and state what time limit it should be provided by (see third bullet of clause W1.3(5)). That time limit is independent of the four weeks stated in clause W1.3(3) if the adjudicator still meets the overall time limit to make its decision, as set out in clause W1.3(8).
The difference is that clause W1.3(3) refers to information which you wish to provide to the adjudicator, whereas clause W1.3(5) refers to any other additional information the adjudicator instructs you to provide.