I am assuming that you are measuring 4.7V across the relay coil terminals, and your supply measures 5.1V.
The transistor will always drop a bit of voltage. A 0.4V drop indicates your transistor is on fairly well, perhaps it could do a bit better.
You have not told us the relay coil resistance, but if I assume 72mA for a JS1A type relay and another 3mA for the LED, that's 75mA. Typically we would like to see base current of about at least 1/20 of that, and you are giving it more than that, so that is fine.
However, your transistor has E and C swapped. What happens in that case, is that the transistor will actually work except its characteristics will mostly be worse. Breakdown voltage will be rated at 5V, not 75V, and current gain will be probably 1/10 or less of what it is when properly connected. Because of the low gain, the Vce ends up being 0.4V when it should be more like 0.1 or 0.2V.