Q: Fuse at hot battery terminal looks burnt out battery keeps loosing charge.

Possibilities include a faulty charging system and/or parasitic power flow, that is current flowing in a circuit, draining the battery, even when the circuit(s) has been commanded off. This is referred to as a "parasitic" load, sometimes known as a phantom load, too. The cost to resolve such an electrical fault, including diagnosing the charging system if a fault exist there, is made up of diagnostic time and the cost of materials or components, if the the parasitic power draw is due to a failed electronic module, relay and so forth. If the electrical fault causing current to flow is pinpointed quickly and readily it should not be too costly to fix. The mechanic would first test the charging system and then look to see if there is any current draw above a threshold (varies by car, but threshold could be 20 to 50 milliamps) when the car is off. If there is an excess load above the threshold when the car is off, with the factory wiring diagram in hand, a Mechanic would use an ammeter to test the circuit, segment by segment, until the fault is found. If you want the required diagnostic steps performed by a certified Mechanic, dispatched by YourMechanic right to your location, please request an electrical diagnostic and the responding certified mechanic will get the issue you are writing in about resolved for you. If you have further questions or concerns, do not hesitate to re-contact YourMechanic as we are always here to help you.