Electronic – How to autoregulate a TP4056 for maximum solar power extraction

The linked PDF specification file that you have shown in "specs", is clearly saying: TP4056 chip itself (NOT THE TP4056 BASED CHARGER-MODULE WHEN PLUGGED IN WITH LI-ION BATTERY) needs minimum 55µA (micro-Ampere) or higher to operate/charge a regular condition Li-ion battery.
''(by the way, usually specs have multiple pages in range of 2 to 10+, but your linked location has only 2-pages)''

1000µA (micro-Ampere) = 1mA (milli-Ampere)

So your source, which is your SOLAR PANEL UNIT (you said it can output 4.5 volt 45mA), must be able to provide near or more-than 55µA current. And according to your answer, your solar panel unit appears to be able to do that.

When sun-light strength is lower, then output current in source will also drop, so you have to consider adding more Solar Panel Units in parallel, if you want to charge battery in lower sun-light condition.

There are other circuit (electronic) components in TP4056 board, & there are (electronic) circuit boards (and chemical components) inside the Li-ion battery itself, etc, these have various types of limitations. Different (electronic & chemical) component needs different level of minimum current & voltage, for it to work/operate . So whichever electronic (or chemical) component will have higher level of minimum-requirement in a same circuit board , that will be the ultimate lowest minimum-requirement to function, in a practical real world circuit/board. Environmental elements (Temperature, etc) also affects operating minimum-requirement.

There are multiple other CHIP(s) based Charger-Modules, you can also choose a suitable another charger module, (if you don't have access to more Solar Panel Units, to increase source/charging current).

Based on shown charts/data inside specs page, it appears, the TP4056 chip based Charger-Module circuit board that the Manufacturer has used, that board itself needed minimum 80-to-100mA current to operate/charge Li-ion battery, so my guess is, your Solar Panel Unit actually supplying near to 80-to-100mA current when sun-light is stronger, so that is one of the reason why your Solar Panel Unit is still able to (slowly) charge your Li-ion battery.

TP4056A appears to be able to operate/charge at minimum 50mA current.

So your power source, which is your SOLAR PANEL UNITs, must be able to supply at-least 80-to-100mA current, then TP4056 based Charger-Module circuit board will be able to charge Li-ion battery.

There are quite a few variations of Charger-Module, that are based on TP4056 chip(s). Multiple manufacturers are making multiple different Charger-Modules based on different versions/editions of 4056, 4056A, etc chip(s). So there may be some 4056/4056A chip based charger-module boards , which may allow charging Li-ion or Li-ion-Polymer (LiPo) based battery at much lower level current than typical 80-to-100mA.

If a Li-ion or Li-ion-Polymer (LiPo) battery is deeply discharged, then there is a trickle charge mode, which needs minimum 120mA current charging. TP4056 chip itself can handle that mode of charging.

CV = constant-voltage . CC = constant-current.

Charging a Li-ion/Li-ion-Polymer battery's CELL (battery-cells are inside the battery) happens with two stages/modes: (1) CC, & then (2) CV.
Charging a Li-ion/Li-ion-Polymer BATTERY (it contains multiple battery-cells) happens with three stages: (1) CC, (2) Balance, & then (3) CV.
There are various (chemical, electro-chemical, etc) material based multiple different types of Li-ion or Li-ion-Polymer (LiPo), etc based batteries, so their (electro-chemical) behaviors are not same. So their charging & discharging etc properties are slightly different.
Different manufacturer uses different types of electronic (protection) circuits for their Li-ion batteries. Some model (aka: variation, version, edition) of battery does not use/have any (protection) circuits.

If you change R_prog & put/use higher value resistor (electronic component), then charging current will be forced to use a lower current/ampere rate. So when supply/source power can give more current/ampere, for-example when sun-light is (very) strong, and at that point of time/moments, supplied current into battery will not be properly utilized, as circuit will attempt to regulate & keep it at lower rate. When sufficient power source is not available to do CC & CV mode of charging, then charging mode will be paused/standby. So a balanced or a variable rate is needed when charging from a variable power output based power-supply (Solar/Wind, etc based). So source power output's intensity level based charging is better.

Larger size capacitor (and 1N58xx schottky diode or similar performing alternatives, etc) can help to store extra power supply from Solar Panel Unit (or Wind Turbine), so that Charger-Module board can utilize that stored power to charge Li-ion battery, when sun-light (or wind/air flow) strength/intensity is lower.

There are also other chip or micro-controller based charger-module that are specifically better at charging from Solar/Wind based power input/source, where power source current (or voltage) intensity can be variable/changing.

This is several questions and should probably be broken apart, but nevertheless

1. According to this TP4056 datasheet, a 1.2 kΩ R_PROG will give a charging current between 950 and 1050 mA. This is a bit much for a lithium-ion battery, which typically prefers a charging current no more than 1 C (e.g. 900 mA for a 900 mAh battery). If you have a datasheet for your battery, it should say the recommended maximum charging current. You might want to go to 0.5 C (450 mA) to reduce battery aging. A 1.43 kΩ resistor would give a current below 900 mA, and a 3 kΩ resistor would give one below 450 mA.

2. The output of a switching regulator like that is typically fairly well-regulated (hence the name “regulator”). There are some graphs of the line and load regulation on this MT3608 datasheet.

3. You almost certainly do not need or want a linear regulator on the output of the boost regulator, as it would reduce the efficiency and therefore battery life a lot. The only common reason to add a linear regulator to the output of a boost regulator is to reduce the output noise, but the only really noise-sensitive component you mentioned is the XBee, and it will need a little 3.3 V regulator anyway (and feel free to use a linear regulator for that).

4. It looks like that USB port is directly connected to the Vin hole, so you can't connect USB power while a battery is connected, or you stand a good chance of exploding the battery. USB is already 5 V nominal, though, so you don't need to connect it to the input of the boost regulator—you can just disconnect the boost regulator and hook USB power up directly to your circuit.