I just got some interesting results.
I was running a breadboard project (prototype of a binary clock) with 6 LEDs switching on and off constantly, maybe an average of 10% LED usage. An Arduino Nano, an RTC and a 74HC595 shift register.
I connected a nearly fully charged 18650 (4.1V) directly to the breadboard and left it running to see how long it would go for.
A week later (last night) the bits for the rest of the proposed power supplied arrived.
A TP4056 + 5V boost module.
So I checked how the battery was doing after the week, 3.56V. So it was doing fine.
I decided as 3.56V was getting awfully close to when the Nano would brown out and crash I would add the TP4056 and Boost module to see how long it would continue running.
It was dead this morning. So it ran for a week on 4.1V - 3.56V, but lasted only 9 hours with the modules in place.
Even if you include inefficiencies of the boost converter that performance stinks! The boost converter was claimed to work down to 2V, but the TP4056 would cut off at around 2.5V.
Of course this make the concept of the power supply rubbish and I would be better off running it direct from an 18650 which I guesstimate would run it for 2 weeks.
I think before we look too deep about efficiency... Let me first understand...
From your description, I assume you are running it like this:
battery -> booster -> TP4056 -> Nano
This is not a very good layout. TP4056 is a charge-management IC. It makes no sense to supply it with the battery.
Each step you are loosing power. So, from battery (3.x to 4.2 volt) you first boost to 5V (efficiency lost), then feed it into to TP4056 which regulates it back down to charge voltage (4.2 - 3.x volt), and if you tap the battery into NANO Vin, you have yet another regulator which drops the voltage a bit further.
Each step you are loosing power. The worst lost is at the boost stage - even if component is working at 100% efficiency, for the same amperage drawn,
boosting from 4V to 5V means you lost a quarter of your power right there. Since the battery operation is working well for you, there is no reason to first boost it. With that in mind, you should reconsider how you stage each module.
If you want to charge (that's what the TP4056 is designed for), it would be silly to run the TP4056 via the battery.
So, the running would be:
Battery-->NANO +5V (bypass the voltage regulator in NANO at Vin)
or if you want +5V for the NANO
Battery-->Boost-->NANO +5V
(Warning, I am a hobbyist with limited experience, but this is the way I would try)If you want to charge with the TP4056 while you run, you need an external source of 7.5V ish.
+7.5V --> TP4056-in
+7.5V --> NANO Vin (not NANO +5V) using NANO's voltage reg to bring 7.5V to 5V.
plus
TP4056-out --> battery
TP4056-out --> NANO USB Vin (which has a power selector diode)
When 7.5V external power is ON, NANO is powered from Vin, and TP4056 is powered. Battery is charging via TP4056-out. TP4056-out also connects to the NANO USB Vin giving it up to 4.2V max. But the USB Vin has a "power selector diode". So, it will draw power from the 5V regulator and the diode also prevents current to flow from NANO's 5V back to battery.
When 7.5V external power is OFF, NANO is powered via the USB's Vin which is your battery (connected to TP4056-out).
Since there are a ton of NANO variations out there, I'd experiment with it first. There may be NANO's out there without the "power-selector diode". Run it with TP4056 powered but without battery to make sure that the TP4056-out (USB Vin) is sitting at or below 4.2V. Connect the battery and make sure it is <= 4.2V, and it terminates charge when full. Otherwise, you would be flow charging the battery which would not be good.
Note that your voltage at the NANO's +5V is really 5V
ONLY when the 7.5V source is ON. When it is off, the NANO's +5V is your battery voltage after the diode drop, which would be at best close to 4.2V. Inserting your Boost module there (between battery and NANO USB in) would mean that the TP4056-out would be connected to both the battery and boost module. That could
mess up charge management a little or a lot.
It mess up by a lot when your
boosted voltage exceeds the NANO's regulator output voltage. If (say for example) your boost output is 5.1V into the NANO's USB-Vin and your NANO's Vin regulator output is lower at just 5.0V, the "power selector diode" is not blocking and power is therefore coming from USB-Vin (ie: from battery). So, the TP4056 is actually powering the battery charging and the NANO plus breadboard at the same time. You will likely be drawing enough current for the TP4056
never to reach full-charge-shutdown. So your battery would be float charging until 7.5V source is off. To prevent that, your boost must be lower than the NANO's 5V regulator output. I would prefer not to boost at all (clean), but if I must, I would set boost to about 4.5V ish to make sure that boost-out voltage is always less than NANO's 5V regulator-out. Then, I am sure when the 7.5V external is ON, my boost circuitry is mere idling supplying no power to the NANO. The TP4056 terminates at about 10% charge current, so when your boost circuitry is merely idling, the current draw may be low enough (less than 1/10 C) for the TP4056 to decide "charge full". You need to experiment with it well. Float charging will kill your battery for sure.
Good luck.