Therefore usually aggrieved party would wish to rescind. But the rights given to a party under a contract may be wider than the object of the venture requires them to be. It may then be worth a party’s while to keep the contract alive for an ancillary purpose and compensate himself by damages for the loss of his main object.
Per Denning LJ in Frederick E. Rose (London) Ltd v William H. Pirn Jnr & Co Ltd  2 QB 450, at p.461:
Rectification is concerned with contracts and documents, not with intentions. In order to get rectification it is necessary to show that the parties were in complete agreement on the terms of their contract, but by an error wrote them down wrongly; and in this regard, in order to ascertain the terms of their contract, you do not look into the inner minds of the parties - into their intentions - any more than you do in the formation of any other contract. You look at their outward acts, that is, at what they said or wrote to one another in coming to their agreement, and then compare it with the document which they have signed. If you can predicate with certainty what their contract was, and that it is, by a common mistake, wrongly expressed in the document, then you rectify the document; but nothing less will suffice.
In Agip SpA v Navigazione Alta Italia SpA (The Nai Genova),  1 Lloyd’s Rep 353, Slade LJ expressed his opinion on the matter of granting rectification in case of unilateral mistake at 365:
…I strongly incline to the view that in the absence of estoppel, fraud, undue influence or a fiduciary relationship between the parties, the authorities do not in any circumstances permit the rectification of a contract on the grounds of unilateral mistake, unless the defendant had actual knowledge of the existence of the relevant mistaken belief at the time when the mistaken plaintiff signed the contract. In view of the drastic nature of such an order, so far as the non-mistaken defendant is concerned, the consequences of any such conclusion may not appear unduly harsh. I do not say that even where estoppel, fraud, undue influence or a fiduciary relationship exists rectification will necessarily be an available or appropriate remedy.
Commission for the New Towns v Cooper (GB) Ltd  2 All ER 929, per Stuart-Smith LJ at pp.943, 946:
‘The commonest circumstance in which rectification is granted is where the written contract does not accurately record the parties' joint agreement. In other words there is a mistake common to both parties. In the case of unilateral mistake, that is to say where only one party is mistaken as to the meaning of the contract, rectification is not ordinarily appropriate. This follows from the ordinary rule that it is the objective intention of the parties which determines the construction of the contract and not the subjective intention of one of them. Also, it would generally be inequitable to compel the other party to execute a contract which he had no intention of making, simply to accord with the mistaken interpretation of the other party…
… But were it necessary to do so in this case, I would hold that where A intends B to be mistaken as to the construction of the agreement, so conducts himself that he diverts B’s attention from discovering the mistake by making false and misleading statements, and B in fact makes the very mistake that A intends, then notwithstanding that A does not actually know, but merely suspects that B is mistaken, and it cannot be shown that the mistake was induced by any misrepresentation, rectification may be granted. A’s conduct is unconscionable and he cannot insist on performance in accordance to the strict letter of the contract; that is sufficient for rescission.