Circuit works on breadboard but has power problems on protoboard
I have a page about power saving.

To summarize some of the techniques you can use:


Run the processor at a lower frequency
Run the processor at a lower voltage
Turn off unneeded internal modules in software (eg. SPI, I2C, Serial, ADC)
Turn off brownout detection
Turn off the Analog-to-Digital converter (ADC)
Turn off the watchdog timer
Put the processor to sleep
Don't use inefficient voltage regulators - if possible run directly from batteries
Don't use power-hungry displays (eg. indicator LEDs, backlit LCDs)
Arrange to wake the processor from sleep only when needed
Turn off (with a MOSFET) external devices (eg. SD cards, temperature sensors) until needed


I've got a temperature and humidity sensor which has been running for years on 3 x AA batteries.




  The circuit is powered by a 9V battery (6LR61) connected to RAW (Vin on other Arduino boards).


That's not a great choice for a start, as the voltage regulator has to turn the 9V into 5V (in the form of wasted heat). You'd be better off with 3 x AA batteries (around 4.5V) and put them straight into the 5V pin.

Those 9V batteries have notoriously low mAh because they are basically 6 coin cells stacked on top of each other.