The mentioned supplication is not amongst the supplications that are prescribed to be recited at the mentioned time. There are particular supplications that are specified to be recited in the early morning in the book "Mafateeh al-Jinan".
The wife deserves the dower as agreed. If the agreement was that it is to be paid at the time of the contract, then it must be given then and when the wife asks for it. If they agree on delaying the payment, then it is permissible to delay it.