OK, the water must be coming from somewhere:
Water from a leak. If you've been checking your meter and don't think you have any leaks inside that's good, but it's possible there is a crack or break in service line upstream of your meter. This might actually explain reduced water pressure in the house. If you suspect this is the problem I would call your water company and have them take a look.
Rainwater. If the water appears only during and shortly after a storm, then you probably have a drainage problem. Make sure all gutters are directed away from the house. You might also look at grading near the house to make sure water will flow away from the house. Is the porch at the bottom of a slope towards the house?
Groundwater. I would be a little surprised if this were groundwater welling up, but it's not impossible. Have you had a very wet spring that would've raised the water table level? Or done any serious landscaping / excavation / compacting in your area?
Septic system. An unhappy septic system can cause water to well up, although probably not so close to the house or clear / odorless. Do you have a septic system?
(I wouldn't beat yourself up too much about not sealing that gap. If water is pooling from below I doubt a little sealant would hold it off for long.)
This part of the foundation is most likely spreading out a roof load from a sizable portion of the sunroom roof. If the soil supporting the foundation has eroded away, this will need to be rectified some time next year.
For now, get a bag of ready mix mortar, the kind you just add water to, it has sand and cement already mixed in in the correct proportion. Mix up a small amount, keep the water to a minimum so it will stay where you put it and not slump down. Patch the hole. That's all this is right now, a temporary patch.
If the hole opens up beyond such that you feel like it's sort of a bottomless pit, cut a short length of something like a paint stick. Thin enough to push through the hole but long enough to wedge inside when turned 90 degrees. Tie a length of wire to the center so you can keep it pulled tight when you push mortar up against it.
As you suspect, water and freeze/thaw cycles chip away at the foundation over time. There was probably once a innocuous hairline crack there at one point. Water got in there and froze, opening the crack. Repeat dozens of times a winter and over the years you get what you see now.
The solution is keep water away from the foundation. I see you have a gutter system over this area. That is good. The patio sloping towards the house is really bad. Not only does it get into foundation cracks and erode the material, water gets under the house structure and can cause uneven settlement, stressing members in ways that was never intended. Some people think crooked floors in old houses are "charming". In reality it is a sign of an unstable structure. Even if the house stays stable, water can be the source of toxic mold. Water has no place in your house other than inside plumbing.
You need to correct the drainage issue. Ideally, it should slope by gravity away from the house at least with a 2% slope. Another solution, not nearly as good, is to place a sump at the lowest point and pump collected water away. I can't emphasize enough how a natural gravity solution is far superior to a mechanical solution. Sometimes you do what you have to do though.
Combine this with the possibility of needing to correct erosion under the foundation. You my be digging up the patio anyway. OTOH, if the foundation is still properly supported and you correct the drainage issue, your little mortar patch could possibly be considered a permanent fix.
Best Answer
An 1/8 inch crack that is weeping water is a fairly serious fault. Wondering, is the crack displaced or flat? Regardless, any crack that allows water to seep through means the crack if through the entire wall. Normally I would recommend chiseling a "key" or an inverted "V" into the concrete and patching it with expanding hydrolic cement and coating the crack with a waterproofing rated for up to 4 PSI. I personally have never used the product you linked to. What is important is that bonds soundly. I would think it might be worth a try since you intend to do a better fix from the exterior in the spring. Remember, a simple concrete pad ove the area will not fix your problem, the leak is ground water, not a little ice melting on the top of the ground.