I can see you use Office 2013 so you could try this one (in for example T3 and copy down):

=IFERROR(AVERAGEIFS($A$3:$A$15,$B$3:$B$15,B3,$A$3: $A$15,"<>"),"")



And yes, you can use wildcards for client names. For example

=IFERROR(AVERAGEIFS($A$3:$A$15,$B$3:$B$15,"O*",$A$ 3:$A$15,"<>"),"")

will return 0.831429 which is the average for all clients where name starts with O.

As stated here the formula cannot be copied to the right without manual adjustment of the references because of the client columns.