Does a bishop have the extraordinary power to allow marriage between first cousins? As per canon law 1091 it is not possible.
Please give me a clear answer.
Can a bishop allow Marriage between cousins?
The short answer is yes: first cousins can marry if the diocesan bishop grants an appropriate dispensation.
To understand the question at hand let us start by looking at what Canon Law has to say about "Specific Diriment Impediments" to marriage.
Can. 1091 §1. In the direct line of consanguinity marriage is invalid between all ancestors and descendants, both legitimate and natural.
§2. In the collateral line marriage is invalid up to and including the fourth degree.
§3. The impediment of consanguinity is not multiplied.
§4. A marriage is never permitted if doubt exists whether the partners are related by consanguinity in any degree of the direct line or in the second degree of the collateral line.
Now let's look at how to interpret Canon 1091.
Canon 1091: CONSANGUINITY: The impediment arises from a blood relationship, either fullblood or half-blood; therefore, its existence does not derive from the marital status of the common ancestor. In the direct line, all marriages are invalid; no dispensation is ever granted from consanguinity in the second degree of the collateral line [brother and sister] (Canon 1078). In the collateral line marriages are invalid to the third degree [aunts or uncles with nephews or nieces] as well as to the fourth degree [first cousins]. This impediment is of ecclesiastical law, and the Local Ordinary or his delegate can dispense for a sufficient reason; however, if civil legislation will not permit such a union, the Local Ordinary will not grant a dispensation (confer Canon 1071, §1, 2/). Beyond the fourth degree of the collateral line, no impediment exists. - Manual of Tribunal Marriage Procedures
From the article:
The church's position is that [two first cousins] are forbidden to marry only by ecclesiastical law, not by divine law. For this reason it is canonically possible to receive a dispensation that permits two first-cousins to validly marry in the Catholic church. (This assumes, of course, that it is legal under civil law in the area in which the marriage is to take place).
While a dioscesan bishop may certainly grant a dispensation allowing two first cousins to marry, rest assured that he will take a good look at any other relations between the two families' bloodlines first.