How to append the dataframe data into dedicated SQL pool through Spark pool of synapse?

The synapsesql.read method (described here) does not currently support appending to a dedicated SQL pool table but you can simulate that behaviour with either partition switching or use CTAS to combine your original table and the new data.

Partitioning pattern:

CTAS pattern:

CREATE TABLE dbo.yourNewTable
WITH (
    DISTRIBUTION = ROUND_ROBIN, -- or HASH(someColumn)|REPLCIATE
    CLUSTERED COLUMNSTORE INDEX -- or CLUSTERED INDEX|HEAP
)
AS
SELECT *
FROM dbo.someTable_part1
UNION ALL
SELECT *
FROM dbo.someTable_part2;

You may be able to do append with a JDBC connection but I probably would not do that in Synapse. You could also try saveAsTable method, bearing in mind this appends to a Spark table not a dedicated SQL pool table, something like:

df.write.saveAsTable('yourTable', mode='append')