Can a bishop allow Marriage between cousins? 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




Canon law relationship chart