Normally, with Linux and BOINC, just starting the Manager causes the client to start. It sounds like it is running correctly from your Start Menu selection. Does Mint have .desktop files? I would think so since is it Debian under the covers just like Ubuntu. How about creating a .desktop file for Boinc Manager. That is one of the things I had to do when I transitioned from 16.04 to 18.04, I found that in 18.04 the BOINC desktop file was not created. So I made one so that I can Dock the Manager for easy access. I don't have it autostart. This is my Boinc Manager desktop file. Don't know if this might help you. If you have a desktop file, it is just like any other desktop file you have in /usr/share/applications and can be simply added to Startup Applications. You will have to adjust directory locations for your apps.

[Desktop Entry]
Encoding=UTF-8
Version=1.0
Type=Application
Name=BOINC Manager
Icon=/home/keith/Desktop/BOINC/boincmgr.32x32.png
Path=/home/keith/Desktop/BOINC
Exec=/home/keith/Desktop/BOINC/boincmgr
StartupNotify=false
StartupWMClass=Boincmgr
OnlyShowIn=Unity;
X-UnityGenerated=true

Now I wonder, if a desktop file is the same as the desklets available for Mint? Sorry, perhaps more a question for Less Bayliss.
Anyway I'll try and try...

[Desktop Entry]
Type=Application
Version=1.0
Exec=boincmgr
Path=/var/lib/boinc-client
Icon=boincmgr
Categories=System;Monitor;GTK;
Keywords=computation;science;
Name=BOINC Manager
GenericName=BOINC monitor and control utility
GenericName[cs]=Monitorovací a ovládací nástroj pro BOINC
GenericName[de]=BOINC Ãœberwachungs- und Kontrollprogramm
GenericName[pt]=Monitorização BOINC e utilitário de controlo
GenericName[pt_BR]=Utilitário de controle e monitoramento do BOINC
Comment=Configure or monitor a BOINC core client
Comment[cs]=Monitoruje a nastavuje klienta BOINC
Comment[de]=BOINC Basis Client konfigurieren oder überwachen
Comment[pt]=Configurar ou monitorizar o cliente básico do BOINC
Comment[pt_BR]=Configure ou monitore o cliente básico do BOINC

Normally, with Linux and BOINC, just starting the Manager causes the client to start. It sounds like it is running correctly from your Start Menu selection. Does Mint have .desktop files? I would think so since is it Debian under the covers just like Ubuntu. How about creating a .desktop file for Boinc Manager. That is one of the things I had to do when I transitioned from 16.04 to 18.04, I found that in 18.04 the BOINC desktop file was not created. So I made one so that I can Dock the Manager for easy access. I don't have it autostart. This is my Boinc Manager desktop file. Don't know if this might help you. If you have a desktop file, it is just like any other desktop file you have in /usr/share/applications and can be simply added to Startup Applications. You will have to adjust directory locations for your apps.

[Desktop Entry]
Encoding=UTF-8
Version=1.0
Type=Application
Name=BOINC Manager
Icon=/home/keith/Desktop/BOINC/boincmgr.32x32.png
Path=/home/keith/Desktop/BOINC
Exec=/home/keith/Desktop/BOINC/boincmgr
StartupNotify=false
StartupWMClass=Boincmgr
OnlyShowIn=Unity;
X-UnityGenerated=true

Don't know anything about Mint . . . . does it have something similar to Startup Applications like Ubuntu? If it does you can pass the startup of the client to the command shell via something like:

sh -c '/usr/bin/boinc'

It sounds like you already have a desktop object to start the Manager and the only thing wrong is the client is not started before invoking the manager. You can always start them manually in a Terminal session to prove things out first. Open a root Terminal session in /usr/bin and type ./boinc to get the client started. Then open another root Terminal session and start the Manager. ./boincmgr and you can see the log of the startup of the client in the Terminal window. If everything works, just get the client running via a Startup Applications entry or make a script to start it via /etc/init.d.

What does the Event Log say at line 4?
That's where you can tell if it's running as a daemon.

And just to be picky, the latest version of Mint is 19.3, Tricia
(Which I don't like as much as the version 18s.)

so when I select manually in the startup menu the boinc-client in the init.d folder this should (hopefully) work?

I think so, it has always started by default on my Ubuntu machines. Certainly when I wanted to be able to run the windows version under WINE removing it stopped the problems of it starting up and preventing the windows client from running.

Never used Mint but I would start by looking in /var/lib/boinc which is the default location.
Have you checked to see if the client is running? Try running

sudo/etc/init.d/boinc-client start

and see if this gets it going.

Thanks. In the location there's a Boinc and Boinc client folder. anyway (sorry I quite new with Linux and still think in Windows patterns) which is the executable file then?
When I start Boinc Manager manually from the start menu everything works and projects are loaded and computed. But the automatic start of the manager fails.

I think you need to make sure the client is one of the programs that runs at start up. On my Ubuntu box that is under setttings>session and startup which is slightly easier than editing the init file. (In my opinion.)

Never used Mint but I would start by looking in /var/lib/boinc which is the default location.
Have you checked to see if the client is running? Try running

sudo/etc/init.d/boinc-client start

and see if this gets it going.