Mrs. Pumphrey, the owner of Tricki, believes that Tricki has a problem with malnutrition.

Explanation:
- In the story "The Dear Departed" by Stanley Houghton, Tricki is a pet dog owned by Mrs. Pumphrey. Mrs. Pumphrey is a wealthy and doting owner who spoils Tricki with extravagant treats and excessive amounts of food.
- Mrs. Pumphrey notices that Tricki has been gaining weight and becoming lethargic. Tricki's excessive weight gain and low energy levels are clear indicators that something is wrong with his health.
- Mrs. Pumphrey consults her friend, Dr. Herriot, who is a veterinarian. Dr. Herriot examines Tricki and concludes that the dog is suffering from malnutrition despite his overweight appearance.
- Malnutrition is a condition where the body lacks essential nutrients, even if the individual is consuming an excess of calories. In Tricki's case, the excessive amounts of rich food that Mrs. Pumphrey has been feeding him are lacking in important vitamins, minerals, and other essential nutrients.
- The excessive treats and indulgent diet have caused Tricki to become overweight, but his body is still lacking the necessary nutrients for proper health. This is why Mrs. Pumphrey believes that malnutrition is the problem affecting Tricki.
- Dr. Herriot advises Mrs. Pumphrey to reduce Tricki's food intake and provide him with a balanced diet to improve his health. However, Mrs. Pumphrey finds it difficult to resist Tricki's pleading eyes and continues to spoil him, exacerbating his health problems.
- The story highlights the consequences of overindulgence and the importance of providing a balanced and nutritious diet for pets. Tricki's malnutrition is a result of Mrs. Pumphrey's misguided affection and lack of understanding about proper pet care.

In conclusion, Mrs. Pumphrey believes that Tricki has a problem with malnutrition due to his excessive weight gain and low energy levels. This highlights the importance of providing pets with a balanced and nutritious diet for their overall well-being.