Does 'poor' reception deplete the battery quicker? In the area where I live I sometimes have trouble with reception. The Internet stops working for me or it takes time to load pages. The phone tries and tries to connect.

It seems to me that when this is happening it depletes the battery much more quickly. Is my observation correct?
Yes, it does cost you more power.

The transceiver circuit is engineered with power saving in mind and will reduce sending power as much as possible if the reception is good. This also reduces the SAR value which is a measurement for exposure of the human body to radiation.

If the reception is bad sending signal strength has to be adjusted accordingly.

Additionally:
The underlying protocols are also designed to retransmit packages in case of a corrupted transmission. This might have an extra impact on your battery too.