Describe the bug

When mounting sse_app() from FastMCP with a URL prefix using Starlette’s Mount, the SSE stream still returns the default /messages/ endpoint without the prefix. This causes the MCP client to resolve an incorrect URL (e.g., /messages/ instead of /mcp/messages/), resulting in a 404 error.

To Reproduce

Steps to reproduce the behavior:

1. Define an MCP server as below:

from mcp.server.fastmcp import FastMCP
from starlette.applications import Starlette
from starlette.routing import Mount

mcp = FastMCP(
    # sse_path="/sse",  # default
    # message_path="/messages/",  # default
)


application = Starlette(
    routes=[
        Mount("/mcp", app=mcp.sse_app()),  # HERE !!!
    ]
)

1. Start the server and navigate to http://127.0.0.1:8000/mcp/sse in your browser
2. Observe the response in the browser. You will receive an SSE event like:

event: endpoint
data: /messages/?session_id=...

At this point, the MCP client performs a urljoin operation between the SSE URL (http://127.0.0.1:8000/mcp/sse) and the endpoint path (/messages/). This causes the resolved endpoint URL to become http://127.0.0.1:8000/messages/.

1. The client tries to connect to http://127.0.0.1:8000/messages/, which results in a 404.

Expected behavior

The SSE stream should return the correct full path reflecting the prefix, e.g.:

data: /mcp/messages/?session_id=...

This would allow the client to connect to the actual valid message endpoint.

Screenshots
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Desktop (please complete the following information):

• OS: macOS
• Browser : Arc
• Version : 1.87.1 (60573)

Smartphone (please complete the following information):

• Device: N/A
• OS: N/A
• Browser : N/A
• Version : N/A

Additional context

This behavior seems to originate from hardcoded endpoint generation inside sse_app():

It would be great to have support for specifying a prefix in sse_app() or for the prefix to be auto-detected from the ASGI scope.

Thank you!