What is the ruling on disposing of money obtained as compensation for an advertisement that was published in a distorted manner or at an untimely date, and should it be returned to the newspaper or to the advertising company, and is it permissible to appoint an agent to repay this money to another person?

Your mentioned work is an agency for a fee, and an agency for a fee has the ruling of an ijara (lease/hire contract). In contracts, what matters are the contents, not the names. It is stated in Al-Taj wa Al-Iklil that an agency with compensation is an ijara that binds both parties and is only permissible with a named wage, a fixed term, and a known work. Therefore, it has the ruling of ijara contracts. The first ruling of ijara is that the benefit and the wage must be known.

If the company hires you for a known benefit, you must perform the work according to the condition. If a significant defect is found in the advertisement after its publication, the hirer is not obligated to pay the full wage, and a deduction is made proportionate to the defect. The value of the advertisement is assessed both in its sound and defective state, and a percentage of the wage is deducted accordingly. The amount refunded by the newspaper is rightfully yours, as long as you have reached an agreement with the company and the value of the defect has been deducted from your funds.