I'm a novice working on a relative's system so I don't have a lot of info to provide. The problem is a gas furnace which cycles on too frequently and often - but not always - blows for only 10 sec. or so. There is also an outdoor mounted A/C unit connected, but I believe it was added some years after the furnace. My initial suspect was a problem with the thermostat - a basic mechanical one, not the original but I don't know much more than that. There is no heat pump. Both the heat and A/C work but the constant heat cycling on and off just seems wrong and wasteful.
Below is the wiring hook-up. I can kind of get that maybe it's a work-around for a jumper between the Rh and Rc connectors. The short white wire from the circuit board to the Y connector still totally baffles me though.
My question is: Can anyone decipher this wiring arrangement and tell me how I can shift over to a 4 wire or 5 wire replacement thermostat?
Thanks for any help you can provide.

The blue wire on the RH terminal - Where does that go? Check the low voltage connections at the furnace. If both red and blue are connected to R terminal, this should be pretty easy. Check and post back.

I'm not familiar with the wiring on a furnace. These are the two obvious wiring junctions but they don't mean much to me. Perhaps you can make some sense of them?

Is this perhaps what you meant by blue and red sharing a common connection?

I'm not familiar with the wiring on a furnace. These are the two obvious wiring junctions but they don't mean much to me. Perhaps you can make some sense of them?

In the picture that shows the limit and fan switch, your fan off tempature is set too high, it should be around 105 or so. To set this unplug the unit from the power. Hold the silver disk and move the pointer that is set at about 150 to where it points to about 105. You may have to adjust this a couple of times before you get it right because this is operated by heat and they get old.

Also do not let the rotating silver disk hit the black wire nut with the white wires in it. Move the wire nut over so it will not hit the silver disk.

In the picture that shows the limit and fan switch, your fan off tempature is set too high, it should be around 105 or so. To set this unplug the unit from the power. Hold the silver disk and move the pointer that is set at about 150 to where it points to about 105. You may have to adjust this a couple of times before you get it right because this is operated by heat and they get old.

I'm not too clear on this. Up close (pic) I can see 3 settings pointers. It looks like moving the middle one closer to 100 makes some sense but 105 would put it right next to the other pointer, which doesn't seem to make sense. As I said I'm new to all this so kindly offer any more advice possible on exactly what adjustments are recommended. Thanks.

You have been given some very good advise. The bottom line is this, if you are not familiar or comfortable with this equipment then you shouldn't be tinkering around with it. Perhaps you should call for on-site service before you cause any damage to the equipment or worse, to yourself.