I think the issue is that your csi  storageclass that supports snapshots is Storage Classes: csi-hostpath-sc but the workload that you are trying to snapshot is using “- name: storageClassName value: local-path-kasten”

If this is k3s then the local-path sc does not support snapshotting. So your application should be using the csi-hostpath-sc and then you should be able to snapshot without issues.

when i use pod with volume class csi-hostpath-sc i get this error

- cause:
    cause:
      fields:
        - name: pvcName
          value: example-pvc-4
        - name: pvName
          value: example-pv-3
        - name: storageClassName
          value: csi-hostpath-sc
        - name: namespace
          value: it
      file: kasten.io/k10/kio/exec/phases/phase/snapshot.go:243
      function: kasten.io/k10/kio/exec/phases/phase.FetchSnapshotSession
      linenumber: 243
      message: Volume Snapshots are not supported for this storage provider.Try K10's
        Generic Storage Backup
        method(https://docs.kasten.io/latest/install/generic.html?#generic-storage-backup-and-restore)
        or contact contact@kasten.io
    file: kasten.io/k10/kio/exec/phases/backup/snapshot_data_phase.go:148
    function: kasten.io/k10/kio/exec/phases/backup.(*SnapshotDataPhase).Run
    linenumber: 148
    message: Failed to fetch the snapshot session
  message: Job failed to be executed
- cause:
    cause:
      fields:
        - name: pvcName
          value: example-pvc-4
        - name: pvName
          value: example-pv-3
        - name: storageClassName
          value: csi-hostpath-sc
        - name: namespace
          value: it
      file: kasten.io/k10/kio/exec/phases/phase/snapshot.go:243
      function: kasten.io/k10/kio/exec/phases/phase.FetchSnapshotSession
      linenumber: 243
      message: Volume Snapshots are not supported for this storage provider.Try K10's
        Generic Storage Backup
        method(https://docs.kasten.io/latest/install/generic.html?#generic-storage-backup-and-restore)
        or contact contact@kasten.io
    file: kasten.io/k10/kio/exec/phases/backup/snapshot_data_phase.go:148
    function: kasten.io/k10/kio/exec/phases/backup.(*SnapshotDataPhase).Run
    linenumber: 148
    message: Failed to fetch the snapshot session
  message: Job failed to be executed

 kubectl get storageclasses  csi-hostpath-sc -o json

{
    "allowVolumeExpansion": true,
    "apiVersion": "storage.k8s.io/v1",
    "kind": "StorageClass",
    "metadata": {
        "annotations": {
            "kubectl.kubernetes.io/last-applied-configuration": "{\"allowVolumeExpansion\":true,\"apiVersion\":\"storage.k8s.io/v1\",\"kind\":\"StorageClass\",\"metadata\":{\"annotations\":{},\"name\":\"csi-hostpath-sc\"},\"provisioner\":\"hostpath.csi.k8s.io\",\"reclaimPolicy\":\"Delete\",\"volumeBindingMode\":\"Immediate\"}\n",
            "storageclass.kubernetes.io/is-default-class": "true"
        },
        "creationTimestamp": "2023-11-29T09:29:44Z",
        "name": "csi-hostpath-sc",
        "resourceVersion": "4616421",
        "uid": "d56ed44f-4058-46ee-bf6f-4eada73ad184"
    },
    "provisioner": "hostpath.csi.k8s.io",
    "reclaimPolicy": "Delete",
    "volumeBindingMode": "Immediate"
}

kubectl version:
GitVersion:"v1.27.3"

kubectl get volumesnapshotclass -o json

{
    "apiVersion": "v1",
    "items":  
        {
            "apiVersion": "snapshot.storage.k8s.io/v1",
            "deletionPolicy": "Delete",
            "driver": "hostpath.csi.k8s.io",
            "kind": "VolumeSnapshotClass",
            "metadata": {
                "annotations": {
                    "k10.kasten.io/is-snapshot-class": "true",
                    "kubectl.kubernetes.io/last-applied-configuration": "{\"apiVersion\":\"snapshot.storage.k8s.io/v1\",\"deletionPolicy\":\"Delete\",\"driver\":\"hostpath.csi.k8s.io\",\"kind\":\"VolumeSnapshotClass\",\"metadata\":{\"annotations\":{\"snapshot.storage.kubernetes.io/is-default-class\":\"true\"},\"name\":\"csi-hostpath-snapclass\"},\"parameters\":null}\n",
                    "snapshot.storage.kubernetes.io/is-default-class": "true"
                },
                "creationTimestamp": "2023-11-29T11:11:35Z",
                "generation": 1,
                "name": "csi-hostpath-snapclass",
                "resourceVersion": "9434560",
                "uid": "86bf9b21-2167-4275-9cb7-9d5284f5c57e"
            }
        }
    ],
    "kind": "List",
    "metadata": {
        "resourceVersion": ""
    }
}

Can you try creating a snapshot outside of Kasten.  https://kubernetes.io/docs/concepts/storage/volume-snapshots/

 created without errors

apiVersion: v1
kind: PersistentVolumeClaim
metadata:
  name: csi-pvc
spec:
  accessModes:
  - ReadWriteOnce
  resources:
    requests:
      storage: 1Gi
  storageClassName: csi-hostpath-sc
---- 
apiVersion: snapshot.storage.k8s.io/v1
kind: VolumeSnapshot
metadata:
  name: new-snapshot-demo
spec:
  volumeSnapshotClassName: csi-hostpath-snapclass
  source:
    persistentVolumeClaimName: csi-pvc

kubectl get  volumesnapshot

NAME                READYTOUSE   SOURCEPVC   SOURCESNAPSHOTCONTENT   RESTORESIZE   SNAPSHOTCLASS            SNAPSHOTCONTENT                                    CREATIONTIME   AGE
new-snapshot-demo   true         csi-pvc                             1Gi           csi-hostpath-snapclass   snapcontent-3525a074-fad5-4ef8-bef3-1504cc65604e   2m15s          4m16s

Hello,

A CSI driver that has Volume Snapshot support. Please look at the list of CSI drivers to confirm snapshot support.

Thank you,

Haythem