How is the antenna return loss measurement impacted by the cable's return loss?
TL;DR - The cable won't have any affect on the measurement if you take the measurement properly.
If you have a VNA, you should be calibrating it before taking a measurement. Typically done with a SOLT (Short, Open, Load, Thru) calibration kit for coaxial cables (and just SOL for a 1-port measurement). The VNA measures the response of the standards in the calibration kit, and uses them to determine characteristics of the &quot;stuff&quot; between the VNA and the device you'll be testing. These characteristics are typically constants determined by various measurements and used in a 12-term error model. Once the calibration measurements are done, the calculated error terms allow the VNA to mathematically remove the effects of your connectors, cables, etc., from the measurement of your actual device.
With a modern VNA with incredibly high dynamic range (on the order of 90+ dB!), even a &quot;terrible&quot; cable with 1dB loss at 2.4GHz will have no effect on an accurate measurement
Now, of course, I agree with @Marcus in the comments to just get a better cable. But that said, you'll still need to calibrate to get a proper measurement. See edit below.
You can search using keywords like &quot;12-term error model/correction&quot; and &quot;VNA calibration&quot; to read more about this. It's a very fundamental and extremely important concept in microwave metrology. Even if you have a VNA that doesn't allow calibration (which I'd be.... very surprised if it was the case), you can take measurements of standards and calculate your own error terms, then measure your device and do the math to &quot;de-embed&quot; the raw data down to the reference plane of your device. Of course, you'll need to get a calibration kit if you don't already have one, but a VNA is all but useless without a calibration kit.

Edit:
On further discussion in the comments, I now understand what you're trying to do. The &quot;cable&quot; you linked in the original question is probably more accurately considered a probe specifically intended for in-situ measurement of your circuit. Your connector is a switched connector that provides isolation from the RF circuit and the antenna when the probe is connected to it, facilitating in-situ measurement. It seems Murata also provides Calibration Adapters which adapt the board connector to SMA, which would allow you to use your SOL standards to be connected to the connector end of the probe and calibrate to that reference plane. Unfortunately, it looks like that would still introduce a little bit of error (since the calibration would be at the SMA reference plane, not at the probe connector), but that's likely a non-issue if you're only interested in the magnitude of \$S_{11}\$ (i.e. the VSWR) and not the phase.