It is pity you could not have come in here and asked for advice prior to purchase because there is little folks like myself can do to help you with 10 days left before your trip. But I'll try to help.
Anker do sell a mains charger, they are under the Powerport name.
Solar (in general, not specific to Anker's products) delivers usually about half what is claimed for a 15Watt I'd assume 7W (its been measured at 7.2W) so usually in perfect conditions I'd expect about 1.6A 5V. In less than perfect conditions the volts and the amps drops. If it drops too far then it ceases being able to be above the threshold of whatever it is output to. Different devices have different thresholds so some are more tolerant of less than perfect conditions than others. So the larger the solar panel, the less often it drops its output below the threshold of the device, i.e. larger panels are useless less often.
So the smaller the solar panel, the more often it is useless. As shown here
The Anker 15W panel is one of the higher rates in reviews.
In the case of a Powercore 20100, I don't know what is its threshold, the minimum voltage which would cause it to recharge, but that is my 1st thought.
You can have 3 items at fault here:
1) The cable. This is the single most common cause of problems. Use a meter to check your cable. In my own tests across a range of cables about 1:4 were faulty. By faulty I mean it would look like they were working but when I put a meter on what should have been about 2A is more like 0.1-0.6A. So I'd 1st check your cable. A meter is best, otherwise try different cables. Without a meter you're mostly running blind.
2) The solar panel. There are plenty of places for a solar panel to fault. Without a meter you cannot easily eliminate this cause. You could connect say a mobile phone or tablet to it and then look at the rate of recharge and infer if the mobile/tablet is charging. Example calculation: a mobile with a 3000mah battery, the "15W" solar panel would give out about 1.7A i.e. 1700mah, so I'd expect 3000/17000 = 1.8 hours = 105 mins, so roughly 1% of charge increase per minute. Note: a mobile being used will show a lot less than this so put leave mobile alone and in shade and come to it in say 30 mins and you should see 30% increase. Also note that all devices trickle charge from 85% to 100% so you need a mostly empty flat mobile like 10% and see it move to 40% in 30 mins, once you get to 85% charged it will then take roughly twice as long go 85% to 100% as it did from 70% to 85%.
3) The Powercore. A meter would help. Show on mains power (wall socket charger) it does actually take a charge and then discharge into something you can measure, e.g. in the example of a 3000mah mobile, the 20100 should recharge the mobile (20100/3000/5*3.7) 5 times.
The main thrust of your text is showing the realization about the core problem with solar, that is mostly useless for most problems relative to a charged powerbank. Solar makes little point on size and weight reasons on shorter trips, and becomes less useful the smaller they get. The next model up from Anker is 21W.
21W is 42% more power for 32% more weight, because the solar part becomes a larger component of the rest of the system. Also the voltage drop off is less. It is the voltage drop-off which is the most critical part, if it drops below the charging threshold it is literally then useless, in fact less than useless because some devices will detect a voltage, open their charging circuits and then not get enough input to offset the charging circuit and so causes the device to discharge rather than charge.
On a size/weight basis, assuming everything worked perfectly, the Powercore 20100 would hold 12 hours of sunshine. ( 20100/1700 = 11.8) It weighs 12.6oz. The Anker 15W weighs about the same (right? Put these on scales). In perfect conditions you'd need 2-3 days of sunshine to get the same energy as the 20100. Do you need 2-3 days worth of 20100 charge more than the 20100 can carry in itself? Well mostly not, you'd need to be off-grid for a week or so before solar makes any sense. And then if you were off-grid for >1 week you'd be better off with a bigger solar panel on a Watts per weight basis. You'd be better off with a group of people and between the group having a bigger solar panel, then the sunshine is much less likely to drop below the threshold voltage of your devices, i.e. it is useless less often.
Note: Apple products and solar in general do not get along. This is because in most cases once the Apple product witnesses a drop off of input power it locks itself at that lower level even when the power input goes up, so you have to disconnect and reconnect the cable if say a cloud passes over.
Important: batteries do not like heat. The sun makes things hot. Solar panels drop off efficiency with heat. Solar in general likes UV, human skin does not like UV. So basically if you literally plugged in a Powercore 20100 into the Anker Solar Lite 15W the heat of the sun drops the 15W claimed to more like 7W as it warms up, the Powercore 20100 would also begin to get warm and that would accelerate its aging, on too a long trip the heat of the sun would fry your Powercore 20100. So then you'd need to place the solar panel in as cool a spot as possible to maximise its efficiency, so propped up say off a surface so it has air circulating, or resting on a light colored surface, and the Powercore or other device out behind something in the shade. Then, the lower the USB cable the more the losses in the cable, so then you'd need a thicker cable with less losses, which are heavy.
Why did you buy the Powercore 20100? You describe the use of tearooms and mains power whilst there, well then you're looking for fastest recharge. The 20100 has one of the worst recharge / mah of the Anker product range. Two Powercore 10000 would recharge twice as fast in total than one 20100. Or the Powercore II 20000, or the Powercore 26800, or.... (QC or USB-C, etc).
- it is a pity you didn't seek more advice earlier, but given where you are now: test and eliminate cause (cable, solar, Powercore).
- buy the Powerport2 and two Powercore 10000. These would collectively recharge your Powercore twice as fast when you do come across mains power. Two Powercore 1000 weigh the same as one Powercore 20100, and recharge twice as fast, if one of them fails you still have the other. The Powerport2 outputs 4.8A which is enough for full speed recharge of two Powercore 10000.
- Assuming nothing is at fault, the Anker Solar Lite 15W should be close in performance to what a mobile phone is capable of ingesting anyway, consider charging your devices directly off solar, it will eliminate the power losses of an intermediate device, as per my calculation examples you should be able to recharge most mobiles fully in about 2 hours of sunshine.