Python 3.10, 58 bytes

lambda m,i=0:sum(r[i:=i-1]+r[~i]*(~i!=len(m)+i)for r in m)

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Python 3, 62 bytes

lambda m:sum(r[i]+r[~i]*(i!=len(m)+~i)for i,r in enumerate(m)) 

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Explanation

lambda m: for i,r in enumerate(m) # iterate m with i = row index and r = row +r[~i]*(i!=len(m)+~i) # add r[len(m) - i - 1] if the index does not equal i r[i] # add r[i] sum( ) # sum numbers generated by the loop