The comptroller may require any person presenting for settlement an account or claim, except a claim with regard to excise and non-property taxes, for any cause against the city or the board of education, to be sworn before the comptroller, any of the deputy comptrollers, or any officer or employee of the comptroller’s office or of the law department designated in a written instrument by the comptroller and filed in the comptroller’s office, touching such account or claim, and when so sworn, to answer orally as to any facts relative to the justness of such account or claim. Wilful false swearing before the comptroller, deputy comptroller or officer or employee designated to conduct such oral examination is perjury and punishable as such. In adjusting and settling such claims, the comptroller, as far as practicable, shall be governed by the rules of law and principles of equity which prevail in courts of justice. Claims against the city or against any of the counties contained within its territorial limits, or payable in the first instance from moneys in the city treasury for services rendered or work done or materials or supplies furnished, except:

1. claims reduced to judgment, or

2. awards, costs, charges and expenses duly taxed or ordered paid in judicial proceedings, or

3. claims arising under the provisions of contracts made at public letting in the manner provided by chapter thirteen of the charter and chapter one of title six of the code, or

4. claims settled and adjusted by the comptroller, pursuant to the authority of this section,

shall not be paid unless an auditor of accounts shall certify that the charges therefor are just and reasonable.

b. Except as hereinbefore otherwise provided, all contracts with the city or any of such counties or with any public officer acting in its or their behalf, shall be subject to audit by the comptroller. The power hereby given to settle and adjust such claims shall not be construed to authorize the comptroller to dispute the amount of any salary established by or under the authority of any officer or department authorized to establish the same, nor to question the due performance of duties by such officer, except when necessary to prevent fraud. If in any action at law against the city to recover upon a claim not embraced within the exceptions specified in subdivision a the amount claimed by the plaintiff is in excess of the amount so audited and settled by the comptroller, the plaintiff must establish a claim by competent evidence of value, and no testimony shall be admitted to show a promise or agreement by any officer or employee of the city or of any of the counties contained within its territorial limits to pay any larger sum than the amount so audited or allowed by the comptroller.