Kubernetes add ca certificate to pods' trust root

If you want to bake the cert in at buildtime, edit your Dockerfile adding the commands to copy the cert from the build context and update the trust. You could even add this as a layer to something from docker hub etc.

COPY my-cert.crt /usr/local/share/ca-certificates/RUN update-ca-certificates

If you're trying to update the trust at runtime things get more complicated. I haven't done this myself, but you might be able to create a configMap containing the certificate, mount it into your container at the above path, and then use an entrypoint script to run update-ca-certificates before your main process.

Updated Edit read option 3:

I can think of 3 options to solve your issue if I was in your scenario:
Option 1.) (The only complete solution I can offer, my other solutions are half solutions unfortunately, credit to Paras Patidar/the following site:)
https://medium.com/@paraspatidar/add-ssl-tls-certificate-or-pem-file-to-kubernetes-pod-s-trusted-root-ca-store-7bed5cd683d

1.) Add certificate to config map:lets say your pem file is my-cert.pem
kubectl -n <namespace-for-config-map-optional> create configmap ca-pemstore — from-file=my-cert.pem

2.) Mount configmap as volume to exiting CA root location of container:mount that config map’s file as one to one file relationship in volume mount in directory /etc/ssl/certs/ as file for example

apiVersion: v1 kind: Podmetadata:  name: cacheconnectsamplespec:      containers:      - name: cacheconnectsample        image: cacheconnectsample:v1        volumeMounts:        - name: ca-pemstore          mountPath: /etc/ssl/certs/my-cert.pem          subPath: my-cert.pem          readOnly: false        ports:        - containerPort: 80        command: [ "dotnet" ]        args: [ "cacheconnectsample.dll" ]      volumes:      - name: ca-pemstore        configMap:          name: ca-pemstore

So I believe the idea here is that /etc/ssl/certs/ is the location of tls certs that are trusted by pods, and the subPath method allows you to add a file without wiping out the contents of the folder, which would contain the k8s secrets.
If all pods share this mountPath, then you might be able to add a pod present and configmap to every namespace, but that's in alpha and is only helpful for static namespaces. (but if this were true then all your pods would trust that cert.)

Option 2.) (Half solution/idea + doesn't exactly answer your question but solves your problem, I'm fairly confident will work in theory, that will require research on your part, but I think you'll find it's the best option:)
In theory you should be able to leverage cert-manager + external-dns + Lets Encrypt Free + a public domain name to replace the self signed cert with a Public Cert.
(there's cert-manager's end result is to auto gen a k8s tls secret signed by Lets Encrypt Free in your cluster, they have a dns01 challenge that can be used to prove you own the cert, which means that you should be able to leverage that solution even without an ingress / even if the cluster is only meant for private network.)

Edit: Option 3.) (After gaining more hands on experience with Kubernetes)
I believe that switchboard.op's answer is probably the best/should be the accepted answer. This "can" be done at runtime, but I'd argue that it should never be done at runtime, doing it at runtime is super hacky and full of edge cases/there's not a universal solution way of doing it.

Also it turns out that my Option 1 doing it is only half correct.mounting the ca.crt on the pod alone isn't enough. After that file is mounted on the pod you'd need to run a command to trust it. And that means you probably need to override the pods startup command. Example you can't do something like connect to database (the default startup command) and then update trusted CA Certs's command. You'd have to override the startup file to be a hand jammed, overwrite the default startup script, update trusted CA Certs's, connect to the database. And the problem is Ubuntu, RHEL, Alpine, and others have different locations where you have to mount the CA Cert's and sometimes different commands to trust the CA Certs so a universal at runtime solution that you can apply to all pods in the cluster to update their ca.cert isn't impossible, but would require tons of if statements and mutating webhooks/complexity. (a hand crafted per pod solution is very possible though if you just need to be able to dynamically update it for a single pod.)

switchboard.op's answer is the way I'd do it if I had to do it. Build a new custom docker image with your custom ca.cert being trusted baked into the image. This is a universal solution, and greatly simplifies the YAML side. And it's relatively easy to do on the docker image side.