It's floating point cancellation, we can use arbitrary precision or exact values:
In[7]:= (E^k k^(-1/a) (a k^(1/a) - Gamma[1/a] + Gamma[1/a, k]))/(
a (-1 + E^k)) /. {a -> 1/10, k -> 1/10}
Out[7]= (100000000000 (-(36287999999999999/100000000000) +
401044422751291/(1000000000 E^(1/10))) E^(1/10))/(-1 + E^(1/10))
In[8]:= N[%, 20]
Out[8]= 0.91283470967626027646
In[9]:= N[%7]
Out[9]= 0.
