MySQL update query locks table for insertion - mysql

Question

I am in a problematic situation and found dozens of questions on same topic, but may b i am not able to understand those solutions as per my issue.
I have a system built in Codeigniter, and it does the following
codeigniter->start_transaction()
UPDATE T SET A = 1, MODIFIED = NOW()
WHERE PK IN
( SELECT PK FROM
(SELECT PK, LAST_INSERT_ID(PK) FROM T
where FK = 31 AND A=0 AND R=1 AND R_FK = 21
AND DEAD = 0 LIMIT 0,1) AS TBL1
) and A=0 AND R = 1 AND R_FK = 21 AND DEAD = 0
-- what this query does is , it takes a row dynamically which is not dead yet,
--and not assigned and it's linked to 21 id (R_FK) from R table,
-- when finds the row, update it to be marked as assigned (A=1).
-- PK = LAST_INSERT_ID(PK) ensures that last_insert_id is updated with this row id, so i can retrieve it from PHP
GOTO MODULE B
MODULE B {
INSERT INTO T(A,B,C,D,E,F,R,FK,R_FK,DEAD,MODIFIED) VALUES(ALL VALUES)
-- this line gives me lock wait timeout exceeded.
}
MySQL version is 5.1.63-community-log
Table T is an INNODB table and has only one normal type index on FK field, and no foreign key constraints are there. PrimaryKey (PK) field is an auto_increment field.
I get lock wait timeout in the above case , and that is due to first transactional update holding lock on table, how can i avoid lock on table with that update query ,while using transactions, I cannot commit the transaction until i receive response from MODULE B .
I don't have much detailed knowledge about DB and structural things, so please bear with me if i said something not making sense.
--UPDATE--
-- TABLE T Structure
CREATE TABLE `T` (
`PK` int(11) NOT NULL AUTO_INCREMENT,
`FK` int(11) DEFAULT NULL,
`P` varchar(1024) DEFAULT NULL,
`DEAD` tinyint(1) NOT NULL DEFAULT '0',
`A` tinyint(1) NOT NULL DEFAULT '0',
`MODIFIED` datetime DEFAULT NULL,
`R` tinyint(4) NOT NULL DEFAULT '0',
`R_FK` int(11) NOT NULL DEFAULT '0',
PRIMARY KEY (`PK`),
KEY `FK_REFERENCE_54` (`FK`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8;
-- Indexes Information
SHOW INDEX FROM T;
1- Field FK, Cardinality 65 , NULL => Yes , Index_Type => BTRee
2- Field PK, Cardinality 11153, Index_Type => BTRee