Apple recovery mode is displaying an error code -5101F

boot into recovery and launch Terminal

thy these commands, with and without the sudo pre-pended:

TIME:

Before 10.14 Mojave: 

sudo ntpdate -vu time.apple.com

10.14 and later:

sudo sntp -sS time.apple.com

If you have a web browser available you can also use: http://time.is 

that is the most obscure one of those I have seen.

In Internet Recovery, these low negative thousands error messages (between -1000 and -6000) indicate that your provided Wi-Fi Network is inadequate in some fashion, and cannot be used (for this purpose) in its current state.

Common problems include:

Hidden Network-name

required use of a proxy server to get Internet access

required use of a login page to get Internet access

Use of a login server or certificate to get Internet access

Use of PPPoE in the Mac to get Internet access (typically only applies to DSL)

Use of Fixed IP address rather than good old DHCP

Variance of the time by more than five minutes

Failure of Router to provide workable DNS server addresses, or providing 0.0.0.0 [may produce -2002f]

If you have an Ethernet port, you may in some cases be able to move your Mac close to the Router and connect to the Router using Ethernet.

The higher the number, the more obscure the problem.

If your battery was completely depleted, the Date&Time could have reverted to the default, which will not work. There is a terminal command to reset to current Internet Date&Time -- post back for assistance.