Per the schematic, the Vin input on that board (the barrel jack one) goes through a schottky diode to the "5v" rail, schottky diodes have a drop of ~0.3v. ...
Per the schematic, the Vin input on that board (the barrel jack one) goes through a schottky diode to the "5v" rail, schottky diodes have a drop of ~0.3v. So that sounds as expected - this is a pretty common practice; often you can get away without doing anything special (so the 5v rail is 4.7v instead of 5v - if your pot is connected between the "5v" rail and ground, it doesn't matter. Of course, when measuring analog voltages that aren't relative to the power rails, one has to take account of it.
The default reference is Vcc (nominally 5V) so you should read 1023 when the analog input is equal to (or maybe slightly above*) Vcc.If you need better accuracy, you can optionally use an external reference or the internal 1.1V reference (with a voltage divider). The internal reference is stable, but there is a tolerance so you may have to calibrate. * Don't go too far over Vcc as the maximum allowable on an input pin is Vcc+ 0.5V. And, there is a +/- 1 count tolerance so there's a chance of reading 1022 with Vcc applied.
I think the correct solution is to feed your board with 5.36v to get 5v after the diode - assuming you can't find a way to run the sensor off the same "5v" as the board. Particularly when you have a sensor that outputs a voltage proportional to it's supply voltage, you really want the sensor and board measuring the voltage running off the same supply. When there are two independent supplies powering different parts of the same circuit, you get into dangerous territory, because you can damage things if one of them is on, and the other isn't, and the one that is on tries to apply a voltage to a pin on the one that isn't powered. If that situation can happen, you need to take countermeasures (a 10k or 4.7k resistor between them is sufficient for most signals, as this limits the maximum current that could flow into the pin of the unpowered chip, while keeping the impedance low enough that the ADC won't have trouble reading it). This also saves you if the voltages aren't exactly matched (and they never are) - though it doesn't help with the fact that you can't measure the whole range of possible voltages if the thing you're measuring can go slightly higher than the supply voltage on the thing doing the measuring).