The pic below shows the induced electrical response of a Gibson style HB with and without cover, during one of the experiments archived in my data.
Note that the test was done without guitar cable, hence the high pitched resonant peaks, above 7khz on this linear scale. Through a guitar cable, the peaks would be lower pitched.

In a Gibson style HB, the screws coil is in many cases a bit darker and weaker than the one with slugs. EHV (RIP) did describe once the screws coil as a boomer while the slugs coil would be the tweeter.
With a cover, here, this difference is a bit less noticeable (and the Foucault currents due to the cover affect the coil with screws as well). Without cover, the difference between coils becomes way more obvious.
The overall difference between cover and no cover is anyway of 1dB to 1.5dB @ resonant frequencies.
Of course, thie pic would vary with a humbucker whose screws and slugs coils are less distinctively different (such HB's exist too).
As shown by the impulse response screenshot, a cover also makes the attack a wee bit slower and with a tad less amplitude - because of eddy currents, once again.
As stated by our fellow members, the kind of alloy constituting a cover defines its sonic influence, as well as its thickness. There's obviously good and bad covers. Those with chrome slightly rise the inductance, BTW. IOW, they make a pickup a tiny bit stronger with more mids.
For those who would like to go further, let's post a tribute to someone else's work, whose results match those found here:
http://kenwillmott.com/blog/archives/246
http://kenwillmott.com/blog/wp-content/uploads/2016/11/Pickup_Cover_Geometry.pdf
FWIW (my 2 cents).
