Nice puzzle - took a while to work out, but I actually found 2 working solutions:
The differences between both solutions is with the values of A, R and T.
Solution 1:
E = 8
G = 5
H = 1
O = 7
S = 3
W = 9
A = 4
R = 2
T = 6
G R A T E --------> 52468 ( sum = 25)
S H O W S -------> 31793 ( sum = 23)
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E A R T H --------> 84261 ( sum = 21)
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Solution 2:
Same values as above, but with different values for A, R and T:
E = 8
G = 5
H = 1
O = 7
S = 3
W = 9
A = 6
R = 4
T = 2
G R A T E --------> 54628 ( sum = 25)
S H O W S -------> 31793 ( sum = 23)
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E A R T H --------> 86421 ( sum = 21)
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