That device on the apron is a relay. It has 4 wires connected to it.It does not need to be grounded. The solenoid is in the starter.
The 2 large gauge wires are internally connected, when powered up; connecting the battery to the starter.
The 2 smaller gauge wires do the switching.One comes from the ig sw. start position. The second does the grounding; either through the clutch sw if a M/T, or through the nss at the back of the A/T.
As far as circuits go, this one is pretty basic.
One of the smaller 2 are likely the culprit.They are brown and yellow. The terminals on the relay are marked. The Br hooks to to "G", the Y to "I". Many times the conections are loose or corroded. So tighten them up. The yellow comes from the ig.sw. through the bulkhead connector, and another connector at the base of the column.. The brown goes to the nss, or clutch sw.
If it responds to "messing around" at the relay,the issue is probably in the connector at the relay. BTW Ive had sticky,or almost worn out, starter brushes do this; minus the "messing around". Also had it happen with a corroded interface between the starter and the bellhouse, even tight, BION (believe it or not), more than once... Good luck.