Let x be the amount of food Kendall gives to the cat the rest of the day.
The total amount of food Kendall gives the cat every day is 4/5 cup.
Since she feeds it 1/2 cup of food in the evening, the amount of food she gives the cat the rest of the day is 4/5 - 1/2 = 2/10 - 5/10 = -3/10 cup.
Therefore, the amount of food Kendall gives the cat the rest of the day is -3/10 cup. Answer: \boxed{-\frac{3}{10}}.