Add ability to manually adjust the running timer's start time

Allows users to retroactively correct the start time of an ongoing timer
without stopping it. A pencil icon in the timer widget opens an inline
time input pre-filled with the current start time; confirming sends the
new time to the backend which validates it is in the past before persisting.

frontend/src/contexts/TimerContext.tsx

@@ -17,6 +17,7 @@ interface TimerContextType {
   elapsedSeconds: number;
   startTimer: (projectId?: string) => Promise<void>;
   updateTimerProject: (projectId?: string | null) => Promise<void>;
+  updateTimerStartTime: (startTime: string) => Promise<void>;
   stopTimer: (projectId?: string) => Promise<TimeEntry | null>;
 }
 
@@ -102,7 +103,14 @@ export function TimerProvider({ children }: { children: ReactNode }) {
 
   const updateTimerProject = useCallback(
     async (projectId?: string | null) => {
-      await updateMutation.mutateAsync(projectId);
+      await updateMutation.mutateAsync({ projectId });
+    },
+    [updateMutation],
+  );
+
+  const updateTimerStartTime = useCallback(
+    async (startTime: string) => {
+      await updateMutation.mutateAsync({ startTime });
     },
     [updateMutation],
   );
@@ -127,6 +135,7 @@ export function TimerProvider({ children }: { children: ReactNode }) {
         elapsedSeconds,
         startTimer,
         updateTimerProject,
+        updateTimerStartTime,
         stopTimer,
       }}
     >